research article continued fractions of order six and new...
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Research ArticleContinued Fractions of Order Six and New EisensteinSeries Identities
Chandrashekar Adiga A Vanitha and M S Surekha
Department of Studies in Mathematics University of Mysore Manasagangotri Mysore 570 006 India
Correspondence should be addressed to Chandrashekar Adiga cadigamathsuni-mysoreacin
Received 25 February 2014 Revised 25 April 2014 Accepted 7 May 2014 Published 27 May 2014
Academic Editor Ahmed Laghribi
Copyright copy 2014 Chandrashekar Adiga et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We prove two identities for Ramanujanrsquos cubic continued fraction and a continued fraction of Ramanujan which are analogues ofRamanujanrsquos identities for the Rogers-Ramanujan continued fraction We further derive Eisenstein series identities associated withRamanujanrsquos cubic continued fraction and Ramanujanrsquos continued fraction of order six
1 Introduction
Throughout this paper we assume that |119902| lt 1 and for eachpositive integer 119899 we use the standard product notation
(119886 119902)0= 1 (119886 119902)
119899=
119899minus1
prod119895=0
(1 minus 119886119902119895) 119899 ge 1
(119886 119902)infin=
infin
prod119895=0
(1 minus 119886119902119895)
(1)
Srinivasa Ramanujan made some significant contributionsto the theory of continued fraction expansions The mostbeautiful continued fraction expansions can be found inChapters 12 and 16 of his second notebook [1]
The celebrated Rogers-Ramanujan continued fraction isdefined by [2]
119877 (119902) = 11990215
119891 (minus119902 minus1199024)
119891 (minus1199022 minus1199023)=
11990215
1 +119902
1 +1199022
1 +1199023
1 + sdot sdot sdot
(2)
where
119891 (119886 119887) =
infin
sum119899=minusinfin
119886119899(119899+1)2
119887119899(119899minus1)2
= 1 +
infin
sum119899=1
(119886119887)119899(119899minus1)2
(119886119899+ 119887119899)
|119886119887| lt 1
(3)
is Ramanujanrsquos general theta functionRamanujan eventually found several generalizations and
ramifications of 119877(119902)which can be found in his notebooks [1]and ldquolost notebookrdquo [3] Recently Liu [4] and Chan et al [5]have established several new identities associated with theRogers-Ramanujan continued fraction 119877(119902) including Eisen-stein series identities involving 119877(119902)
The beautiful Ramanujanrsquos cubic continued fraction119866(119902)first introduced by Srinivasa Ramanujan in his second letterto Hardy [2 page xxvii] is defined by
119866 (119902) = 11990213
119891 (minus119902 minus1199025)
119891 (minus1199023 minus1199023)
=11990213
1 +119902 + 1199022
1 +1199022+ 1199024
1 +1199023+ 1199026
1 + sdot sdot sdot
(4)
Hindawi Publishing CorporationJournal of NumbersVolume 2014 Article ID 643241 6 pageshttpdxdoiorg1011552014643241
2 Journal of Numbers
Adiga et al [6] Bhargava et al [7] Chan [8] and Vasukiet al [9] have proved several elegant theorems for119866(119902) manyof which are analogues of well-known properties satisfied bythe Rogers-Ramanujan continued fraction
Recently Vasuki et al [10] have studied the followingcontinued fraction of order six
119883(119902) = 11990214
119891 (minus119902 minus1199025)
119891 (minus1199022 minus1199024)
=11990214
(1 minus 119902minus2)
(1 minus 119902minus32) +(11990214
minus 119902minus14
) (119902minus74
minus 11990274)
(1 minus 119902minus32) (1 + 1199023) + sdot sdot sdot
(5)
The continued fraction (5) is a special case of a fascinatingcontinued fraction identity recorded by Ramanujan in hissecond notebook [1] [11 page 24] Furthermore they haveestablished modular relations between the continued frac-tions119883(119902) and119883(119902119899) for 119899 = 2 3 5 7 and 11
In Section 3 of this paper we establish two new identitiesassociated with the continued fractions 119866(119902) and 119883(119902)using the quintuple product identity In Section 4 we deriveEisenstein series identities associated with 119866(119902) and119883(119902)
2 Definitions and Preliminary Results
In this section we present some basic definitions and pre-liminary results One of the most interesting special cases of119891(119886 119887) is [11 Entry 22]
119891 (minus119902) = 119891 (minus119902 minus1199022) =
infin
sum119899=minusinfin
(minus1)119899119902119899(3119899minus1)2
= (119902 119902)infin (6)
Note that the Dedekind eta function 120578(120591) = 119902124
119891(minus119902)where 119902 = 119890
2120587119894120591 Im 120591 gt 0 We need the following threelemmas to prove our main results
Lemma 1 (see [11 Entry 30 page 46]) One has
119891 (119886 119887) + 119891 (minus119886 minus119887) = 2119891 (1198863119887 1198861198873) (7)
Lemma 2 (see [11 page 80]) One has
119891(11986131199021199025
1198613) minus 119861
2119891(
119902
1198613 11986131199025)
= 119891 (minus1199022)119891 (minus119861
2 minus11990221198612)
119891 (119861119902 119902119861)
(8)
Lemma 3 (see [12 Lemma 2(ii)]) Let 119898 = [119904(119904 minus 119903)] 119897 =119898(119904minus119903)minus119903 119896 = minus119898(119904minus119903)+119904 and ℎ = 119898119903minus(119898(119898minus1)(119904minus119903))20 le 119903 lt 119904 Here [119909] denotes the largest integer less than or equalto 119909 Then
(i) 119891(119902minus119903 119902119904) = 119902minusℎ119891(119902119897 119902119896)
(ii) 119891(minus119902minus119903 minus119902119904) = (minus1)119898119902minusℎ119891(minus119902119897 minus119902119896)
3 Main Results
The Jacobi triple product identity states thatinfin
sum119899=minusinfin
(minus1)119899119902119899(119899minus1)2
119911119899= (119902 119902)
infin(119911 119902)infin(119902
119911 119902)infin
119911 = 0
(9)
In Ramanujanrsquos notation the Jacobi triple product identitytakes the form
119891 (119886 119887) = (minus119886 119886119887)infin(minus119887 119886119887)
infin(119886119887 119886119887)
infin (10)
The Jacobi triple product identity was first proved by Gauss[13] Using (3) we have
119891 (minus1198902119894119911 minus119902119890minus2119894119911
) = 2119894119890119894119911
infin
sum119899=0
(minus1)119899+1
119902119899(119899+1)2 sin (2119899 + 1) 119911
(11)
Putting 119886 = minus1198902119894119911 and 119887 = minus119902119890minus2119894119911 in (10) we obtain
119891 (minus1198902119894119911 minus119902119890minus2119894119911
) = (1 minus 1198902119894119911) (1199021198902119894119911 119902)infin(119902119890minus2119894119911
119902)infin(119902 119902)infin
(12)
Putting 119911 = 1205876 and 119911 = 21205876 respectively in (12) we obtain
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
= (1 minus 1198901198941205873
) (1199021198901198941205873
119902)infin(119902119890minus1198941205873
119902)infin(119902 119902)infin
(13)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= (1 minus 11989011989421205873
) (11990211989011989421205873
119902)infin(119902119890minus11989421205873
119902)infin(119902 119902)infin
(14)
Multiplying (13) and (14) together and using the identities
(1 minus 1198901198941205873
) (1 minus 11989011989421205873
) = minus119894radic3
(1 minus 119909) (1 minus 11990911989021205871198946
) (1 minus 119909119890minus21205871198946
) (1 minus 11990911989041205871198946
)
times (1 minus 119909119890minus41205871198946
) (1 minus 11990911989061205871198946
) = (1 minus 1199096)
(15)
in the resulting equation and then after some simplificationswe obtain the following identity
119891 (minus1198901198941205873
minus119902119890minus1198941205873
) 119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= minus119894radic3119902minus14 1205782(120591) 120578 (6120591)
120578 (2120591)
(16)
Theorem 4 Let |119902| lt 1 120572 = minus1 and 120573 = 1 Theninfin
prod119899=1
1
(1 + 1205721199021198992 + 119902119899)minus
infin
prod119899=1
1
(1 + 1205731199021198992 + 119902119899)
= 4119902124 120578 (120591) 120578
2(3120591)
120578 (120591) 1205782 (31205912) + 1205782 (1205912) 120578 (3120591)119866 (119902)
(17)
infin
prod119899=1
1
(1 + 1205721199021198992 + 119902119899)minus
infin
prod119899=1
1
(1 + 1205731199021198992 + 119902119899)
= 4119902124 120578 (120591) 120578 (2120591) 120578 (6120591)
120578 (120591) 1205782 (31205912) + 1205782 (1205912) 120578 (3120591)119883 (119902)
(18)
Journal of Numbers 3
Proof Wemay rewrite (13) and (14) as follows
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
= (1 minus 1198901198941205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= (1 minus 11989011989421205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
(19)
where
120572 = minus2 cos 1205873= minus1 120573 = minus2 cos 2120587
3= 1 (20)
Then
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
(1 minus 1198901198941205873) (21)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
(1 minus 11989011989421205873) (22)
Multiplying the above two equations together and then using(16) in resulting identity we find thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) (1 + 120573119902
119899+ 1199022119899) = 119902minus16 120578 (6120591)
120578 (2120591) (23)
Subtracting (22) from (21) we obtaininfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902124
120578 (120591)(119891 (minus119890
1198941205873 minus119902119890minus1198941205873
)
(1 minus 1198901198941205873)minus119891 (minus119890
11989421205873 minus119902119890minus11989421205873
)
(1 minus 11989011989421205873))
(24)
Using (11) in (24) we deduce thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902minus112
120578 (120591)
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
(25)
where
119875 (119899) =2 sin (2119899 + 1) (1205876)119894119890minus1198941205876 (1 minus 1198901198941205873)
minus2 sin (2119899 + 1) (21205876)119894119890minus11989421205876 (1 minus 11989011989421205873)
(26)
Now by direct computations we find that
119875 (6119898 + 0) = 0 119875 (6119898 + 1) = 2
119875 (6119898 + 2) = 2 119875 (6119898 + 3) = minus2
119875 (6119898 + 4) = minus 2 119875 (6119898 + 5) = 0
(27)
Thereforeinfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2minus
infin
sum119898=0
119902(12119898+3)
28+
infin
sum119898=0
119902(12119898+5)
28
+
infin
sum119898=0
119902(12119898+7)
28minus
infin
sum119898=0
119902(12119898+9)
28
(28)
In the right-hand side of the above equation changing 119898 to119898minus1 in the first two summations and also changing119898 to minus119898in the last two summations we obtaininfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2
infin
sum119898=minusinfin
119902(12119898minus7)
28minus
infin
sum119898=minusinfin
119902(12119898minus9)
28
(29)
Now using the definition of 119891(119886 119887) in the right-hand side ofthe above equation we find that
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= minus211990298
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233)
(30)
In the quintuple product identity (8) replacing 119902 by 1199026 andthen setting 119861 = 119902 we find that
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233) =
119891 (minus11990212) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027) (31)
Combining (25) (30) and (31) we find that
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=minus21199022524
120578 (120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(32)
Dividing both sides of (32) byprodinfin119899=1
(1+120572119902119899+1199022119899)(1+120573119902
119899+1199022119899)
and then using (23) we obtain
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)minus1
minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)minus1
= 21199022924 120578 (2120591)
120578 (120591) 120578 (6120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(33)
Replacing 119886 by 119902 and 119887 by 1199022 in (7) we find that
119891 (119902 1199022) + 119891 (minus119902 minus119902
2) = 2119891 (119902
5 1199027) (34)
4 Journal of Numbers
Now using the above equation in the right-hand side of(33) and then changing 119902 to 11990212 throughout we obtain (17)Equation (18) follows from the following identity
119866 (119902) =120578 (2120591) 120578 (6120591)
1205782 (3120591)119883 (119902) (35)
This completes the proof of Theorem 4
4 Eisenstein Series Identities Associatedwith 119866(119902) and 119883(119902)
In this section we present four Eisenstein series identitiesassociated with 119866(119902) and119883(119902)
Theorem 5 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=1205783(18120591)
120578 (6120591)[
1
119866 (1199023)minus 1 minus 119866 (119902
3)]
(36)
Proof Changing 119899 tominus119899 in the second summation of the left-hand side of Theorem 5 we have
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
minus
minus1
sum119899=minusinfin
119899equiv1(mod 3)
119902minus119899minus 119902minus3119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199023119899+1
1 minus 11990218119899+6minus
infin
sum119899=minusinfin
1199029119899+3
1 minus 11990218119899+6+
infin
sum119899=minusinfin
11990215119899+5
1 minus 11990218119899+6
(37)
Using a corollary of Ramanujanrsquos1Ψ1summation formula [11
Entry 17 page 32]
infin
sum119899=minusinfin
119911119899
1 minus 119886119902119899=(119886119911 119902119886119911 119902 119902 119902)
infin
(119886 119902119886 119911 119902119911 119902)infin
10038161003816100381610038161199021003816100381610038161003816 lt |119911| lt 1 (38)
and Lemma 3 in (37) we find that
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
= 119902(1199029 1199029 11990218 11990218 11990218)infin
(1199026 11990212 1199023 11990215 11990218)infin
minus 1199023(11990215 1199023 11990218 11990218 11990218)infin
(1199026 11990212 1199029 1199029 11990218)infin
+ 1199025(11990221 119902minus3 11990218 11990218 11990218)infin
(1199026 11990212 11990215 1199023 11990218)infin
=(11990218 11990218)2
infin
(1199026 11990212 11990218)infin
119902119891 (minus119902
9 minus1199029)
119891 (minus1199023 minus11990215)minus 1199023119891 (minus119902
3 minus11990215)
119891 (minus1199029 minus1199029)
minus1199022119891 (minus119902
3 minus11990215)
119891 (minus1199023 minus11990215)
(39)
Using (4) in (39) and after some simplifications we obtain(36)
Theorem 6 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=120578 (12120591) 120578 (18120591) 120578 (36120591)
120578 (6120591)
times [1
119883 (1199026)minus
1205782(18120591)
120578 (12120591) 120578 (36120591)minus 119883 (119902
6)]
(40)
Proof Using the identity
infin
sum119899=0
119909119899
1 minus 119910119902119899
=
infin
sum119899=0
infin
sum119898=0
119909119899119910119898119902119898119899
=
infin
sum119899=0
119910119899
1 minus 119909119902119899 |119909|
10038161003816100381610038161199101003816100381610038161003816 lt 1
(41)
the left-hand side of Theorem 6 can be written as
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899
1 minus 1199026119899minus
minus1
sum119899=minusinfin
119899equiv1(mod 6)
minus119902minus2119899
minus 119902minus4119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199026119899+1
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990212119899+2
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990224119899+4
1 minus 11990236119899+6
(42)
Journal of Numbers 5
Using (38) in (42) we obtain
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=(11990236 11990236)2
infin
(1199026 11990230 11990236)infin
119902(11990212 11990224 11990236)infin
(1199026 11990230 11990236)infin
minus 1199022(11990218 11990218 11990236)infin
(11990212 11990224 11990236)infin
minus1199024(1199026 11990230 11990236)infin
(11990212 11990224 11990236)infin
(43)
Using (5) in (43) and after some simplifications we obtain(40)
Differentiating both sides of (12) and then setting 119911 = 0
yield
1198911015840(minus1 minus119902) = minus2119894(119902 119902)
3
infin (44)
where 1198911015840 denotes the partial derivative of 119891with respect to 119911Now we prove a lemma which is useful to prove Eisen-
stein series identities associated with 119866(119902) and119883(119902)
Lemma 7 Consider the following
infin
sum119899=1
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899sin 2119899119911
= (minus11989021198941199111198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023) 119891 (minus119902 minus119902
5)
times119891 (minus1198904119894119911 minus1199026119890minus4119894119911
))
times (41199022119891 (minus119902119890
2119894119911 minus1199025119890minus2119894119911
)
times 119891(minus119902minus11198902119894119911 minus1199027119890minus2119894119911
)119891(minus11990221198902119894119911 minus1199024119890minus2119894119911
))minus1
times1
119891 (minus119902minus21198902119894119911 minus1199028119890minus2119894119911)
(45)
Proof For simplicity we use119865(119911 119902) to denote the logarithmicderivative of 11989411990218119890minus119894119911119891(minus1198902119894119911 minus119902119890minus2119894119911) with respect to 119911 Toprove this lemma we need the following identity which canbe found in [14 Theorem 5] [15 Corollary 2]
119865 (1199111 119902) + 119865 (119911
2 119902) + 119865 (119911
3 119902) minus 119865 (119911
1+ 1199112+ 1199113 119902)
= (minus1198911015840(minus1 minus119902) 119891 (minus119890
2119894(1199111+1199112) minus119902119890minus2119894(1199111+1199112))
times119891 (minus1198902119894(1199112+1199113) minus119902119890minus2119894(1199112+1199113)))
times (119891 (minus11989021198941199111 minus119902119890minus21198941199111)
times 119891(minus11989021198941199112 minus119902119890minus21198941199112)119891(minus119890
21198941199113 minus119902119890minus21198941199113))minus1
times119891 (minus119890
2119894(1199111+1199113) minus119902119890minus2119894(1199111+1199113))
119891 (minus1198902119894(1199111+1199112+1199113) minus119902119890minus2119894(1199111+1199112+1199113))
(46)
As the proof of this lemma is similar to that of Lemma 1 in[4] we omit the details
Using (45) we derive the following Eisenstein seriesidentity
Theorem 8 Let |119902| lt 1 Then
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
=1205786(3120591)
1205782 (120591)119866 (119902) =
1205784(3120591) 120578 (6120591) 120578 (2120591)
1205782 (120591)119883 (119902)
(47)
Proof Dividing both sides of (45) by 119911 and then letting 119911 rarr
0 we obtain
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
= ((1199026 1199026)6
infin119891 (minus119902
3 minus1199023) 119891 (minus119902
1 minus1199025))
times (1199022119891 (minus119902
1 minus1199025) 119891 (minus119902
minus1 minus1199027)
times 119891(minus1199022 minus1199024)119891(minus119902
minus2 minus1199028))minus1
(48)
Using Lemma 3 in (48) we complete the proof ofTheorem 8
We use (sdot119901) to denote the Legendre symbol modulo119901 Setting 119911 = 1205873 in (45) and noting that sin (21198991205873) =
(radic32)(1198993) we find that
radic3
2
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
= (minus11989021198941205873
1198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023)
times119891 (minus1199021 minus1199025) 119891 (minus119890
41198941205873 minus1199026119890minus41198941205873
))
times (41199022119891 (minus119890
2119894((1205873)+120587120591) minus1199026119890minus2119894((1205873)+120587120591)
)
times 119891(minus1198902119894((1205873)minus120587120591)
minus1199026119890minus2119894((1205873)minus120587120591)
))minus1
times (1) (119891 (minus1198902119894((1205873)+2120587120591)
minus1199026119890minus2119894((1205873)+2120587120591)
)
times 119891(minus1198902119894((1205873)minus2120587120591)
minus1199026119890minus2119894((1205873)minus2120587120591)
))minus1
(49)
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Numbers
Adiga et al [6] Bhargava et al [7] Chan [8] and Vasukiet al [9] have proved several elegant theorems for119866(119902) manyof which are analogues of well-known properties satisfied bythe Rogers-Ramanujan continued fraction
Recently Vasuki et al [10] have studied the followingcontinued fraction of order six
119883(119902) = 11990214
119891 (minus119902 minus1199025)
119891 (minus1199022 minus1199024)
=11990214
(1 minus 119902minus2)
(1 minus 119902minus32) +(11990214
minus 119902minus14
) (119902minus74
minus 11990274)
(1 minus 119902minus32) (1 + 1199023) + sdot sdot sdot
(5)
The continued fraction (5) is a special case of a fascinatingcontinued fraction identity recorded by Ramanujan in hissecond notebook [1] [11 page 24] Furthermore they haveestablished modular relations between the continued frac-tions119883(119902) and119883(119902119899) for 119899 = 2 3 5 7 and 11
In Section 3 of this paper we establish two new identitiesassociated with the continued fractions 119866(119902) and 119883(119902)using the quintuple product identity In Section 4 we deriveEisenstein series identities associated with 119866(119902) and119883(119902)
2 Definitions and Preliminary Results
In this section we present some basic definitions and pre-liminary results One of the most interesting special cases of119891(119886 119887) is [11 Entry 22]
119891 (minus119902) = 119891 (minus119902 minus1199022) =
infin
sum119899=minusinfin
(minus1)119899119902119899(3119899minus1)2
= (119902 119902)infin (6)
Note that the Dedekind eta function 120578(120591) = 119902124
119891(minus119902)where 119902 = 119890
2120587119894120591 Im 120591 gt 0 We need the following threelemmas to prove our main results
Lemma 1 (see [11 Entry 30 page 46]) One has
119891 (119886 119887) + 119891 (minus119886 minus119887) = 2119891 (1198863119887 1198861198873) (7)
Lemma 2 (see [11 page 80]) One has
119891(11986131199021199025
1198613) minus 119861
2119891(
119902
1198613 11986131199025)
= 119891 (minus1199022)119891 (minus119861
2 minus11990221198612)
119891 (119861119902 119902119861)
(8)
Lemma 3 (see [12 Lemma 2(ii)]) Let 119898 = [119904(119904 minus 119903)] 119897 =119898(119904minus119903)minus119903 119896 = minus119898(119904minus119903)+119904 and ℎ = 119898119903minus(119898(119898minus1)(119904minus119903))20 le 119903 lt 119904 Here [119909] denotes the largest integer less than or equalto 119909 Then
(i) 119891(119902minus119903 119902119904) = 119902minusℎ119891(119902119897 119902119896)
(ii) 119891(minus119902minus119903 minus119902119904) = (minus1)119898119902minusℎ119891(minus119902119897 minus119902119896)
3 Main Results
The Jacobi triple product identity states thatinfin
sum119899=minusinfin
(minus1)119899119902119899(119899minus1)2
119911119899= (119902 119902)
infin(119911 119902)infin(119902
119911 119902)infin
119911 = 0
(9)
In Ramanujanrsquos notation the Jacobi triple product identitytakes the form
119891 (119886 119887) = (minus119886 119886119887)infin(minus119887 119886119887)
infin(119886119887 119886119887)
infin (10)
The Jacobi triple product identity was first proved by Gauss[13] Using (3) we have
119891 (minus1198902119894119911 minus119902119890minus2119894119911
) = 2119894119890119894119911
infin
sum119899=0
(minus1)119899+1
119902119899(119899+1)2 sin (2119899 + 1) 119911
(11)
Putting 119886 = minus1198902119894119911 and 119887 = minus119902119890minus2119894119911 in (10) we obtain
119891 (minus1198902119894119911 minus119902119890minus2119894119911
) = (1 minus 1198902119894119911) (1199021198902119894119911 119902)infin(119902119890minus2119894119911
119902)infin(119902 119902)infin
(12)
Putting 119911 = 1205876 and 119911 = 21205876 respectively in (12) we obtain
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
= (1 minus 1198901198941205873
) (1199021198901198941205873
119902)infin(119902119890minus1198941205873
119902)infin(119902 119902)infin
(13)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= (1 minus 11989011989421205873
) (11990211989011989421205873
119902)infin(119902119890minus11989421205873
119902)infin(119902 119902)infin
(14)
Multiplying (13) and (14) together and using the identities
(1 minus 1198901198941205873
) (1 minus 11989011989421205873
) = minus119894radic3
(1 minus 119909) (1 minus 11990911989021205871198946
) (1 minus 119909119890minus21205871198946
) (1 minus 11990911989041205871198946
)
times (1 minus 119909119890minus41205871198946
) (1 minus 11990911989061205871198946
) = (1 minus 1199096)
(15)
in the resulting equation and then after some simplificationswe obtain the following identity
119891 (minus1198901198941205873
minus119902119890minus1198941205873
) 119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= minus119894radic3119902minus14 1205782(120591) 120578 (6120591)
120578 (2120591)
(16)
Theorem 4 Let |119902| lt 1 120572 = minus1 and 120573 = 1 Theninfin
prod119899=1
1
(1 + 1205721199021198992 + 119902119899)minus
infin
prod119899=1
1
(1 + 1205731199021198992 + 119902119899)
= 4119902124 120578 (120591) 120578
2(3120591)
120578 (120591) 1205782 (31205912) + 1205782 (1205912) 120578 (3120591)119866 (119902)
(17)
infin
prod119899=1
1
(1 + 1205721199021198992 + 119902119899)minus
infin
prod119899=1
1
(1 + 1205731199021198992 + 119902119899)
= 4119902124 120578 (120591) 120578 (2120591) 120578 (6120591)
120578 (120591) 1205782 (31205912) + 1205782 (1205912) 120578 (3120591)119883 (119902)
(18)
Journal of Numbers 3
Proof Wemay rewrite (13) and (14) as follows
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
= (1 minus 1198901198941205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= (1 minus 11989011989421205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
(19)
where
120572 = minus2 cos 1205873= minus1 120573 = minus2 cos 2120587
3= 1 (20)
Then
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
(1 minus 1198901198941205873) (21)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
(1 minus 11989011989421205873) (22)
Multiplying the above two equations together and then using(16) in resulting identity we find thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) (1 + 120573119902
119899+ 1199022119899) = 119902minus16 120578 (6120591)
120578 (2120591) (23)
Subtracting (22) from (21) we obtaininfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902124
120578 (120591)(119891 (minus119890
1198941205873 minus119902119890minus1198941205873
)
(1 minus 1198901198941205873)minus119891 (minus119890
11989421205873 minus119902119890minus11989421205873
)
(1 minus 11989011989421205873))
(24)
Using (11) in (24) we deduce thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902minus112
120578 (120591)
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
(25)
where
119875 (119899) =2 sin (2119899 + 1) (1205876)119894119890minus1198941205876 (1 minus 1198901198941205873)
minus2 sin (2119899 + 1) (21205876)119894119890minus11989421205876 (1 minus 11989011989421205873)
(26)
Now by direct computations we find that
119875 (6119898 + 0) = 0 119875 (6119898 + 1) = 2
119875 (6119898 + 2) = 2 119875 (6119898 + 3) = minus2
119875 (6119898 + 4) = minus 2 119875 (6119898 + 5) = 0
(27)
Thereforeinfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2minus
infin
sum119898=0
119902(12119898+3)
28+
infin
sum119898=0
119902(12119898+5)
28
+
infin
sum119898=0
119902(12119898+7)
28minus
infin
sum119898=0
119902(12119898+9)
28
(28)
In the right-hand side of the above equation changing 119898 to119898minus1 in the first two summations and also changing119898 to minus119898in the last two summations we obtaininfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2
infin
sum119898=minusinfin
119902(12119898minus7)
28minus
infin
sum119898=minusinfin
119902(12119898minus9)
28
(29)
Now using the definition of 119891(119886 119887) in the right-hand side ofthe above equation we find that
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= minus211990298
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233)
(30)
In the quintuple product identity (8) replacing 119902 by 1199026 andthen setting 119861 = 119902 we find that
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233) =
119891 (minus11990212) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027) (31)
Combining (25) (30) and (31) we find that
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=minus21199022524
120578 (120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(32)
Dividing both sides of (32) byprodinfin119899=1
(1+120572119902119899+1199022119899)(1+120573119902
119899+1199022119899)
and then using (23) we obtain
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)minus1
minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)minus1
= 21199022924 120578 (2120591)
120578 (120591) 120578 (6120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(33)
Replacing 119886 by 119902 and 119887 by 1199022 in (7) we find that
119891 (119902 1199022) + 119891 (minus119902 minus119902
2) = 2119891 (119902
5 1199027) (34)
4 Journal of Numbers
Now using the above equation in the right-hand side of(33) and then changing 119902 to 11990212 throughout we obtain (17)Equation (18) follows from the following identity
119866 (119902) =120578 (2120591) 120578 (6120591)
1205782 (3120591)119883 (119902) (35)
This completes the proof of Theorem 4
4 Eisenstein Series Identities Associatedwith 119866(119902) and 119883(119902)
In this section we present four Eisenstein series identitiesassociated with 119866(119902) and119883(119902)
Theorem 5 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=1205783(18120591)
120578 (6120591)[
1
119866 (1199023)minus 1 minus 119866 (119902
3)]
(36)
Proof Changing 119899 tominus119899 in the second summation of the left-hand side of Theorem 5 we have
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
minus
minus1
sum119899=minusinfin
119899equiv1(mod 3)
119902minus119899minus 119902minus3119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199023119899+1
1 minus 11990218119899+6minus
infin
sum119899=minusinfin
1199029119899+3
1 minus 11990218119899+6+
infin
sum119899=minusinfin
11990215119899+5
1 minus 11990218119899+6
(37)
Using a corollary of Ramanujanrsquos1Ψ1summation formula [11
Entry 17 page 32]
infin
sum119899=minusinfin
119911119899
1 minus 119886119902119899=(119886119911 119902119886119911 119902 119902 119902)
infin
(119886 119902119886 119911 119902119911 119902)infin
10038161003816100381610038161199021003816100381610038161003816 lt |119911| lt 1 (38)
and Lemma 3 in (37) we find that
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
= 119902(1199029 1199029 11990218 11990218 11990218)infin
(1199026 11990212 1199023 11990215 11990218)infin
minus 1199023(11990215 1199023 11990218 11990218 11990218)infin
(1199026 11990212 1199029 1199029 11990218)infin
+ 1199025(11990221 119902minus3 11990218 11990218 11990218)infin
(1199026 11990212 11990215 1199023 11990218)infin
=(11990218 11990218)2
infin
(1199026 11990212 11990218)infin
119902119891 (minus119902
9 minus1199029)
119891 (minus1199023 minus11990215)minus 1199023119891 (minus119902
3 minus11990215)
119891 (minus1199029 minus1199029)
minus1199022119891 (minus119902
3 minus11990215)
119891 (minus1199023 minus11990215)
(39)
Using (4) in (39) and after some simplifications we obtain(36)
Theorem 6 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=120578 (12120591) 120578 (18120591) 120578 (36120591)
120578 (6120591)
times [1
119883 (1199026)minus
1205782(18120591)
120578 (12120591) 120578 (36120591)minus 119883 (119902
6)]
(40)
Proof Using the identity
infin
sum119899=0
119909119899
1 minus 119910119902119899
=
infin
sum119899=0
infin
sum119898=0
119909119899119910119898119902119898119899
=
infin
sum119899=0
119910119899
1 minus 119909119902119899 |119909|
10038161003816100381610038161199101003816100381610038161003816 lt 1
(41)
the left-hand side of Theorem 6 can be written as
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899
1 minus 1199026119899minus
minus1
sum119899=minusinfin
119899equiv1(mod 6)
minus119902minus2119899
minus 119902minus4119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199026119899+1
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990212119899+2
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990224119899+4
1 minus 11990236119899+6
(42)
Journal of Numbers 5
Using (38) in (42) we obtain
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=(11990236 11990236)2
infin
(1199026 11990230 11990236)infin
119902(11990212 11990224 11990236)infin
(1199026 11990230 11990236)infin
minus 1199022(11990218 11990218 11990236)infin
(11990212 11990224 11990236)infin
minus1199024(1199026 11990230 11990236)infin
(11990212 11990224 11990236)infin
(43)
Using (5) in (43) and after some simplifications we obtain(40)
Differentiating both sides of (12) and then setting 119911 = 0
yield
1198911015840(minus1 minus119902) = minus2119894(119902 119902)
3
infin (44)
where 1198911015840 denotes the partial derivative of 119891with respect to 119911Now we prove a lemma which is useful to prove Eisen-
stein series identities associated with 119866(119902) and119883(119902)
Lemma 7 Consider the following
infin
sum119899=1
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899sin 2119899119911
= (minus11989021198941199111198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023) 119891 (minus119902 minus119902
5)
times119891 (minus1198904119894119911 minus1199026119890minus4119894119911
))
times (41199022119891 (minus119902119890
2119894119911 minus1199025119890minus2119894119911
)
times 119891(minus119902minus11198902119894119911 minus1199027119890minus2119894119911
)119891(minus11990221198902119894119911 minus1199024119890minus2119894119911
))minus1
times1
119891 (minus119902minus21198902119894119911 minus1199028119890minus2119894119911)
(45)
Proof For simplicity we use119865(119911 119902) to denote the logarithmicderivative of 11989411990218119890minus119894119911119891(minus1198902119894119911 minus119902119890minus2119894119911) with respect to 119911 Toprove this lemma we need the following identity which canbe found in [14 Theorem 5] [15 Corollary 2]
119865 (1199111 119902) + 119865 (119911
2 119902) + 119865 (119911
3 119902) minus 119865 (119911
1+ 1199112+ 1199113 119902)
= (minus1198911015840(minus1 minus119902) 119891 (minus119890
2119894(1199111+1199112) minus119902119890minus2119894(1199111+1199112))
times119891 (minus1198902119894(1199112+1199113) minus119902119890minus2119894(1199112+1199113)))
times (119891 (minus11989021198941199111 minus119902119890minus21198941199111)
times 119891(minus11989021198941199112 minus119902119890minus21198941199112)119891(minus119890
21198941199113 minus119902119890minus21198941199113))minus1
times119891 (minus119890
2119894(1199111+1199113) minus119902119890minus2119894(1199111+1199113))
119891 (minus1198902119894(1199111+1199112+1199113) minus119902119890minus2119894(1199111+1199112+1199113))
(46)
As the proof of this lemma is similar to that of Lemma 1 in[4] we omit the details
Using (45) we derive the following Eisenstein seriesidentity
Theorem 8 Let |119902| lt 1 Then
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
=1205786(3120591)
1205782 (120591)119866 (119902) =
1205784(3120591) 120578 (6120591) 120578 (2120591)
1205782 (120591)119883 (119902)
(47)
Proof Dividing both sides of (45) by 119911 and then letting 119911 rarr
0 we obtain
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
= ((1199026 1199026)6
infin119891 (minus119902
3 minus1199023) 119891 (minus119902
1 minus1199025))
times (1199022119891 (minus119902
1 minus1199025) 119891 (minus119902
minus1 minus1199027)
times 119891(minus1199022 minus1199024)119891(minus119902
minus2 minus1199028))minus1
(48)
Using Lemma 3 in (48) we complete the proof ofTheorem 8
We use (sdot119901) to denote the Legendre symbol modulo119901 Setting 119911 = 1205873 in (45) and noting that sin (21198991205873) =
(radic32)(1198993) we find that
radic3
2
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
= (minus11989021198941205873
1198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023)
times119891 (minus1199021 minus1199025) 119891 (minus119890
41198941205873 minus1199026119890minus41198941205873
))
times (41199022119891 (minus119890
2119894((1205873)+120587120591) minus1199026119890minus2119894((1205873)+120587120591)
)
times 119891(minus1198902119894((1205873)minus120587120591)
minus1199026119890minus2119894((1205873)minus120587120591)
))minus1
times (1) (119891 (minus1198902119894((1205873)+2120587120591)
minus1199026119890minus2119894((1205873)+2120587120591)
)
times 119891(minus1198902119894((1205873)minus2120587120591)
minus1199026119890minus2119894((1205873)minus2120587120591)
))minus1
(49)
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Numbers 3
Proof Wemay rewrite (13) and (14) as follows
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
= (1 minus 1198901198941205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
= (1 minus 11989011989421205873
) 119902minus124
120578 (120591)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
(19)
where
120572 = minus2 cos 1205873= minus1 120573 = minus2 cos 2120587
3= 1 (20)
Then
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus1198901198941205873
minus119902119890minus1198941205873
)
(1 minus 1198901198941205873) (21)
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899) =
119902124
120578 (120591)
119891 (minus11989011989421205873
minus119902119890minus11989421205873
)
(1 minus 11989011989421205873) (22)
Multiplying the above two equations together and then using(16) in resulting identity we find thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) (1 + 120573119902
119899+ 1199022119899) = 119902minus16 120578 (6120591)
120578 (2120591) (23)
Subtracting (22) from (21) we obtaininfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902124
120578 (120591)(119891 (minus119890
1198941205873 minus119902119890minus1198941205873
)
(1 minus 1198901198941205873)minus119891 (minus119890
11989421205873 minus119902119890minus11989421205873
)
(1 minus 11989011989421205873))
(24)
Using (11) in (24) we deduce thatinfin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=119902minus112
120578 (120591)
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
(25)
where
119875 (119899) =2 sin (2119899 + 1) (1205876)119894119890minus1198941205876 (1 minus 1198901198941205873)
minus2 sin (2119899 + 1) (21205876)119894119890minus11989421205876 (1 minus 11989011989421205873)
(26)
Now by direct computations we find that
119875 (6119898 + 0) = 0 119875 (6119898 + 1) = 2
119875 (6119898 + 2) = 2 119875 (6119898 + 3) = minus2
119875 (6119898 + 4) = minus 2 119875 (6119898 + 5) = 0
(27)
Thereforeinfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2minus
infin
sum119898=0
119902(12119898+3)
28+
infin
sum119898=0
119902(12119898+5)
28
+
infin
sum119898=0
119902(12119898+7)
28minus
infin
sum119898=0
119902(12119898+9)
28
(28)
In the right-hand side of the above equation changing 119898 to119898minus1 in the first two summations and also changing119898 to minus119898in the last two summations we obtaininfin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= 2
infin
sum119898=minusinfin
119902(12119898minus7)
28minus
infin
sum119898=minusinfin
119902(12119898minus9)
28
(29)
Now using the definition of 119891(119886 119887) in the right-hand side ofthe above equation we find that
infin
sum119899=0
(minus1)119899119875 (119899) 119902
(2119899+1)28
= minus211990298
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233)
(30)
In the quintuple product identity (8) replacing 119902 by 1199026 andthen setting 119861 = 119902 we find that
119891 (1199029 11990227) minus 1199022119891 (1199023 11990233) =
119891 (minus11990212) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027) (31)
Combining (25) (30) and (31) we find that
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899) minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)
=minus21199022524
120578 (120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(32)
Dividing both sides of (32) byprodinfin119899=1
(1+120572119902119899+1199022119899)(1+120573119902
119899+1199022119899)
and then using (23) we obtain
infin
prod119899=1
(1 + 120572119902119899+ 1199022119899)minus1
minus
infin
prod119899=1
(1 + 120573119902119899+ 1199022119899)minus1
= 21199022924 120578 (2120591)
120578 (120591) 120578 (6120591)119891 (minus119902
12) 119891 (minus119902
2 minus11990210)
119891 (1199025 1199027)
(33)
Replacing 119886 by 119902 and 119887 by 1199022 in (7) we find that
119891 (119902 1199022) + 119891 (minus119902 minus119902
2) = 2119891 (119902
5 1199027) (34)
4 Journal of Numbers
Now using the above equation in the right-hand side of(33) and then changing 119902 to 11990212 throughout we obtain (17)Equation (18) follows from the following identity
119866 (119902) =120578 (2120591) 120578 (6120591)
1205782 (3120591)119883 (119902) (35)
This completes the proof of Theorem 4
4 Eisenstein Series Identities Associatedwith 119866(119902) and 119883(119902)
In this section we present four Eisenstein series identitiesassociated with 119866(119902) and119883(119902)
Theorem 5 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=1205783(18120591)
120578 (6120591)[
1
119866 (1199023)minus 1 minus 119866 (119902
3)]
(36)
Proof Changing 119899 tominus119899 in the second summation of the left-hand side of Theorem 5 we have
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
minus
minus1
sum119899=minusinfin
119899equiv1(mod 3)
119902minus119899minus 119902minus3119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199023119899+1
1 minus 11990218119899+6minus
infin
sum119899=minusinfin
1199029119899+3
1 minus 11990218119899+6+
infin
sum119899=minusinfin
11990215119899+5
1 minus 11990218119899+6
(37)
Using a corollary of Ramanujanrsquos1Ψ1summation formula [11
Entry 17 page 32]
infin
sum119899=minusinfin
119911119899
1 minus 119886119902119899=(119886119911 119902119886119911 119902 119902 119902)
infin
(119886 119902119886 119911 119902119911 119902)infin
10038161003816100381610038161199021003816100381610038161003816 lt |119911| lt 1 (38)
and Lemma 3 in (37) we find that
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
= 119902(1199029 1199029 11990218 11990218 11990218)infin
(1199026 11990212 1199023 11990215 11990218)infin
minus 1199023(11990215 1199023 11990218 11990218 11990218)infin
(1199026 11990212 1199029 1199029 11990218)infin
+ 1199025(11990221 119902minus3 11990218 11990218 11990218)infin
(1199026 11990212 11990215 1199023 11990218)infin
=(11990218 11990218)2
infin
(1199026 11990212 11990218)infin
119902119891 (minus119902
9 minus1199029)
119891 (minus1199023 minus11990215)minus 1199023119891 (minus119902
3 minus11990215)
119891 (minus1199029 minus1199029)
minus1199022119891 (minus119902
3 minus11990215)
119891 (minus1199023 minus11990215)
(39)
Using (4) in (39) and after some simplifications we obtain(36)
Theorem 6 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=120578 (12120591) 120578 (18120591) 120578 (36120591)
120578 (6120591)
times [1
119883 (1199026)minus
1205782(18120591)
120578 (12120591) 120578 (36120591)minus 119883 (119902
6)]
(40)
Proof Using the identity
infin
sum119899=0
119909119899
1 minus 119910119902119899
=
infin
sum119899=0
infin
sum119898=0
119909119899119910119898119902119898119899
=
infin
sum119899=0
119910119899
1 minus 119909119902119899 |119909|
10038161003816100381610038161199101003816100381610038161003816 lt 1
(41)
the left-hand side of Theorem 6 can be written as
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899
1 minus 1199026119899minus
minus1
sum119899=minusinfin
119899equiv1(mod 6)
minus119902minus2119899
minus 119902minus4119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199026119899+1
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990212119899+2
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990224119899+4
1 minus 11990236119899+6
(42)
Journal of Numbers 5
Using (38) in (42) we obtain
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=(11990236 11990236)2
infin
(1199026 11990230 11990236)infin
119902(11990212 11990224 11990236)infin
(1199026 11990230 11990236)infin
minus 1199022(11990218 11990218 11990236)infin
(11990212 11990224 11990236)infin
minus1199024(1199026 11990230 11990236)infin
(11990212 11990224 11990236)infin
(43)
Using (5) in (43) and after some simplifications we obtain(40)
Differentiating both sides of (12) and then setting 119911 = 0
yield
1198911015840(minus1 minus119902) = minus2119894(119902 119902)
3
infin (44)
where 1198911015840 denotes the partial derivative of 119891with respect to 119911Now we prove a lemma which is useful to prove Eisen-
stein series identities associated with 119866(119902) and119883(119902)
Lemma 7 Consider the following
infin
sum119899=1
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899sin 2119899119911
= (minus11989021198941199111198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023) 119891 (minus119902 minus119902
5)
times119891 (minus1198904119894119911 minus1199026119890minus4119894119911
))
times (41199022119891 (minus119902119890
2119894119911 minus1199025119890minus2119894119911
)
times 119891(minus119902minus11198902119894119911 minus1199027119890minus2119894119911
)119891(minus11990221198902119894119911 minus1199024119890minus2119894119911
))minus1
times1
119891 (minus119902minus21198902119894119911 minus1199028119890minus2119894119911)
(45)
Proof For simplicity we use119865(119911 119902) to denote the logarithmicderivative of 11989411990218119890minus119894119911119891(minus1198902119894119911 minus119902119890minus2119894119911) with respect to 119911 Toprove this lemma we need the following identity which canbe found in [14 Theorem 5] [15 Corollary 2]
119865 (1199111 119902) + 119865 (119911
2 119902) + 119865 (119911
3 119902) minus 119865 (119911
1+ 1199112+ 1199113 119902)
= (minus1198911015840(minus1 minus119902) 119891 (minus119890
2119894(1199111+1199112) minus119902119890minus2119894(1199111+1199112))
times119891 (minus1198902119894(1199112+1199113) minus119902119890minus2119894(1199112+1199113)))
times (119891 (minus11989021198941199111 minus119902119890minus21198941199111)
times 119891(minus11989021198941199112 minus119902119890minus21198941199112)119891(minus119890
21198941199113 minus119902119890minus21198941199113))minus1
times119891 (minus119890
2119894(1199111+1199113) minus119902119890minus2119894(1199111+1199113))
119891 (minus1198902119894(1199111+1199112+1199113) minus119902119890minus2119894(1199111+1199112+1199113))
(46)
As the proof of this lemma is similar to that of Lemma 1 in[4] we omit the details
Using (45) we derive the following Eisenstein seriesidentity
Theorem 8 Let |119902| lt 1 Then
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
=1205786(3120591)
1205782 (120591)119866 (119902) =
1205784(3120591) 120578 (6120591) 120578 (2120591)
1205782 (120591)119883 (119902)
(47)
Proof Dividing both sides of (45) by 119911 and then letting 119911 rarr
0 we obtain
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
= ((1199026 1199026)6
infin119891 (minus119902
3 minus1199023) 119891 (minus119902
1 minus1199025))
times (1199022119891 (minus119902
1 minus1199025) 119891 (minus119902
minus1 minus1199027)
times 119891(minus1199022 minus1199024)119891(minus119902
minus2 minus1199028))minus1
(48)
Using Lemma 3 in (48) we complete the proof ofTheorem 8
We use (sdot119901) to denote the Legendre symbol modulo119901 Setting 119911 = 1205873 in (45) and noting that sin (21198991205873) =
(radic32)(1198993) we find that
radic3
2
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
= (minus11989021198941205873
1198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023)
times119891 (minus1199021 minus1199025) 119891 (minus119890
41198941205873 minus1199026119890minus41198941205873
))
times (41199022119891 (minus119890
2119894((1205873)+120587120591) minus1199026119890minus2119894((1205873)+120587120591)
)
times 119891(minus1198902119894((1205873)minus120587120591)
minus1199026119890minus2119894((1205873)minus120587120591)
))minus1
times (1) (119891 (minus1198902119894((1205873)+2120587120591)
minus1199026119890minus2119894((1205873)+2120587120591)
)
times 119891(minus1198902119894((1205873)minus2120587120591)
minus1199026119890minus2119894((1205873)minus2120587120591)
))minus1
(49)
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Numbers
Now using the above equation in the right-hand side of(33) and then changing 119902 to 11990212 throughout we obtain (17)Equation (18) follows from the following identity
119866 (119902) =120578 (2120591) 120578 (6120591)
1205782 (3120591)119883 (119902) (35)
This completes the proof of Theorem 4
4 Eisenstein Series Identities Associatedwith 119866(119902) and 119883(119902)
In this section we present four Eisenstein series identitiesassociated with 119866(119902) and119883(119902)
Theorem 5 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=1205783(18120591)
120578 (6120591)[
1
119866 (1199023)minus 1 minus 119866 (119902
3)]
(36)
Proof Changing 119899 tominus119899 in the second summation of the left-hand side of Theorem 5 we have
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
=
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
minus
minus1
sum119899=minusinfin
119899equiv1(mod 3)
119902minus119899minus 119902minus3119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199023119899+1
1 minus 11990218119899+6minus
infin
sum119899=minusinfin
1199029119899+3
1 minus 11990218119899+6+
infin
sum119899=minusinfin
11990215119899+5
1 minus 11990218119899+6
(37)
Using a corollary of Ramanujanrsquos1Ψ1summation formula [11
Entry 17 page 32]
infin
sum119899=minusinfin
119911119899
1 minus 119886119902119899=(119886119911 119902119886119911 119902 119902 119902)
infin
(119886 119902119886 119911 119902119911 119902)infin
10038161003816100381610038161199021003816100381610038161003816 lt |119911| lt 1 (38)
and Lemma 3 in (37) we find that
infin
sum119899=1
119899equiv1(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899minus
infin
sum119899=1
119899equiv2(mod 3)
119902119899minus 1199023119899+ 1199025119899
1 minus 1199026119899
= 119902(1199029 1199029 11990218 11990218 11990218)infin
(1199026 11990212 1199023 11990215 11990218)infin
minus 1199023(11990215 1199023 11990218 11990218 11990218)infin
(1199026 11990212 1199029 1199029 11990218)infin
+ 1199025(11990221 119902minus3 11990218 11990218 11990218)infin
(1199026 11990212 11990215 1199023 11990218)infin
=(11990218 11990218)2
infin
(1199026 11990212 11990218)infin
119902119891 (minus119902
9 minus1199029)
119891 (minus1199023 minus11990215)minus 1199023119891 (minus119902
3 minus11990215)
119891 (minus1199029 minus1199029)
minus1199022119891 (minus119902
3 minus11990215)
119891 (minus1199023 minus11990215)
(39)
Using (4) in (39) and after some simplifications we obtain(36)
Theorem 6 Let |119902| lt 1 Then
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=120578 (12120591) 120578 (18120591) 120578 (36120591)
120578 (6120591)
times [1
119883 (1199026)minus
1205782(18120591)
120578 (12120591) 120578 (36120591)minus 119883 (119902
6)]
(40)
Proof Using the identity
infin
sum119899=0
119909119899
1 minus 119910119902119899
=
infin
sum119899=0
infin
sum119898=0
119909119899119910119898119902119898119899
=
infin
sum119899=0
119910119899
1 minus 119909119902119899 |119909|
10038161003816100381610038161199101003816100381610038161003816 lt 1
(41)
the left-hand side of Theorem 6 can be written as
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899
1 minus 1199026119899minus
minus1
sum119899=minusinfin
119899equiv1(mod 6)
minus119902minus2119899
minus 119902minus4119899
+ 119902minus5119899
1 minus 119902minus6119899
=
infin
sum119899=minusinfin
1199026119899+1
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990212119899+2
1 minus 11990236119899+6minus
infin
sum119899=minusinfin
11990224119899+4
1 minus 11990236119899+6
(42)
Journal of Numbers 5
Using (38) in (42) we obtain
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=(11990236 11990236)2
infin
(1199026 11990230 11990236)infin
119902(11990212 11990224 11990236)infin
(1199026 11990230 11990236)infin
minus 1199022(11990218 11990218 11990236)infin
(11990212 11990224 11990236)infin
minus1199024(1199026 11990230 11990236)infin
(11990212 11990224 11990236)infin
(43)
Using (5) in (43) and after some simplifications we obtain(40)
Differentiating both sides of (12) and then setting 119911 = 0
yield
1198911015840(minus1 minus119902) = minus2119894(119902 119902)
3
infin (44)
where 1198911015840 denotes the partial derivative of 119891with respect to 119911Now we prove a lemma which is useful to prove Eisen-
stein series identities associated with 119866(119902) and119883(119902)
Lemma 7 Consider the following
infin
sum119899=1
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899sin 2119899119911
= (minus11989021198941199111198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023) 119891 (minus119902 minus119902
5)
times119891 (minus1198904119894119911 minus1199026119890minus4119894119911
))
times (41199022119891 (minus119902119890
2119894119911 minus1199025119890minus2119894119911
)
times 119891(minus119902minus11198902119894119911 minus1199027119890minus2119894119911
)119891(minus11990221198902119894119911 minus1199024119890minus2119894119911
))minus1
times1
119891 (minus119902minus21198902119894119911 minus1199028119890minus2119894119911)
(45)
Proof For simplicity we use119865(119911 119902) to denote the logarithmicderivative of 11989411990218119890minus119894119911119891(minus1198902119894119911 minus119902119890minus2119894119911) with respect to 119911 Toprove this lemma we need the following identity which canbe found in [14 Theorem 5] [15 Corollary 2]
119865 (1199111 119902) + 119865 (119911
2 119902) + 119865 (119911
3 119902) minus 119865 (119911
1+ 1199112+ 1199113 119902)
= (minus1198911015840(minus1 minus119902) 119891 (minus119890
2119894(1199111+1199112) minus119902119890minus2119894(1199111+1199112))
times119891 (minus1198902119894(1199112+1199113) minus119902119890minus2119894(1199112+1199113)))
times (119891 (minus11989021198941199111 minus119902119890minus21198941199111)
times 119891(minus11989021198941199112 minus119902119890minus21198941199112)119891(minus119890
21198941199113 minus119902119890minus21198941199113))minus1
times119891 (minus119890
2119894(1199111+1199113) minus119902119890minus2119894(1199111+1199113))
119891 (minus1198902119894(1199111+1199112+1199113) minus119902119890minus2119894(1199111+1199112+1199113))
(46)
As the proof of this lemma is similar to that of Lemma 1 in[4] we omit the details
Using (45) we derive the following Eisenstein seriesidentity
Theorem 8 Let |119902| lt 1 Then
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
=1205786(3120591)
1205782 (120591)119866 (119902) =
1205784(3120591) 120578 (6120591) 120578 (2120591)
1205782 (120591)119883 (119902)
(47)
Proof Dividing both sides of (45) by 119911 and then letting 119911 rarr
0 we obtain
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
= ((1199026 1199026)6
infin119891 (minus119902
3 minus1199023) 119891 (minus119902
1 minus1199025))
times (1199022119891 (minus119902
1 minus1199025) 119891 (minus119902
minus1 minus1199027)
times 119891(minus1199022 minus1199024)119891(minus119902
minus2 minus1199028))minus1
(48)
Using Lemma 3 in (48) we complete the proof ofTheorem 8
We use (sdot119901) to denote the Legendre symbol modulo119901 Setting 119911 = 1205873 in (45) and noting that sin (21198991205873) =
(radic32)(1198993) we find that
radic3
2
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
= (minus11989021198941205873
1198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023)
times119891 (minus1199021 minus1199025) 119891 (minus119890
41198941205873 minus1199026119890minus41198941205873
))
times (41199022119891 (minus119890
2119894((1205873)+120587120591) minus1199026119890minus2119894((1205873)+120587120591)
)
times 119891(minus1198902119894((1205873)minus120587120591)
minus1199026119890minus2119894((1205873)minus120587120591)
))minus1
times (1) (119891 (minus1198902119894((1205873)+2120587120591)
minus1199026119890minus2119894((1205873)+2120587120591)
)
times 119891(minus1198902119894((1205873)minus2120587120591)
minus1199026119890minus2119894((1205873)minus2120587120591)
))minus1
(49)
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Numbers 5
Using (38) in (42) we obtain
infin
sum119899=1
119899equiv1(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
minus
infin
sum119899=1
119899equiv5(mod 6)
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=(11990236 11990236)2
infin
(1199026 11990230 11990236)infin
119902(11990212 11990224 11990236)infin
(1199026 11990230 11990236)infin
minus 1199022(11990218 11990218 11990236)infin
(11990212 11990224 11990236)infin
minus1199024(1199026 11990230 11990236)infin
(11990212 11990224 11990236)infin
(43)
Using (5) in (43) and after some simplifications we obtain(40)
Differentiating both sides of (12) and then setting 119911 = 0
yield
1198911015840(minus1 minus119902) = minus2119894(119902 119902)
3
infin (44)
where 1198911015840 denotes the partial derivative of 119891with respect to 119911Now we prove a lemma which is useful to prove Eisen-
stein series identities associated with 119866(119902) and119883(119902)
Lemma 7 Consider the following
infin
sum119899=1
119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899sin 2119899119911
= (minus11989021198941199111198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023) 119891 (minus119902 minus119902
5)
times119891 (minus1198904119894119911 minus1199026119890minus4119894119911
))
times (41199022119891 (minus119902119890
2119894119911 minus1199025119890minus2119894119911
)
times 119891(minus119902minus11198902119894119911 minus1199027119890minus2119894119911
)119891(minus11990221198902119894119911 minus1199024119890minus2119894119911
))minus1
times1
119891 (minus119902minus21198902119894119911 minus1199028119890minus2119894119911)
(45)
Proof For simplicity we use119865(119911 119902) to denote the logarithmicderivative of 11989411990218119890minus119894119911119891(minus1198902119894119911 minus119902119890minus2119894119911) with respect to 119911 Toprove this lemma we need the following identity which canbe found in [14 Theorem 5] [15 Corollary 2]
119865 (1199111 119902) + 119865 (119911
2 119902) + 119865 (119911
3 119902) minus 119865 (119911
1+ 1199112+ 1199113 119902)
= (minus1198911015840(minus1 minus119902) 119891 (minus119890
2119894(1199111+1199112) minus119902119890minus2119894(1199111+1199112))
times119891 (minus1198902119894(1199112+1199113) minus119902119890minus2119894(1199112+1199113)))
times (119891 (minus11989021198941199111 minus119902119890minus21198941199111)
times 119891(minus11989021198941199112 minus119902119890minus21198941199112)119891(minus119890
21198941199113 minus119902119890minus21198941199113))minus1
times119891 (minus119890
2119894(1199111+1199113) minus119902119890minus2119894(1199111+1199113))
119891 (minus1198902119894(1199111+1199112+1199113) minus119902119890minus2119894(1199111+1199112+1199113))
(46)
As the proof of this lemma is similar to that of Lemma 1 in[4] we omit the details
Using (45) we derive the following Eisenstein seriesidentity
Theorem 8 Let |119902| lt 1 Then
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
=1205786(3120591)
1205782 (120591)119866 (119902) =
1205784(3120591) 120578 (6120591) 120578 (2120591)
1205782 (120591)119883 (119902)
(47)
Proof Dividing both sides of (45) by 119911 and then letting 119911 rarr
0 we obtain
infin
sum119899=1
119899 (119902119899minus 1199022119899minus 1199024119899+ 1199025119899)
1 minus 1199026119899
= ((1199026 1199026)6
infin119891 (minus119902
3 minus1199023) 119891 (minus119902
1 minus1199025))
times (1199022119891 (minus119902
1 minus1199025) 119891 (minus119902
minus1 minus1199027)
times 119891(minus1199022 minus1199024)119891(minus119902
minus2 minus1199028))minus1
(48)
Using Lemma 3 in (48) we complete the proof ofTheorem 8
We use (sdot119901) to denote the Legendre symbol modulo119901 Setting 119911 = 1205873 in (45) and noting that sin (21198991205873) =
(radic32)(1198993) we find that
radic3
2
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
= (minus11989021198941205873
1198911015840(minus1 minus119902
6) 119891 (minus119902
3 minus1199023)
times119891 (minus1199021 minus1199025) 119891 (minus119890
41198941205873 minus1199026119890minus41198941205873
))
times (41199022119891 (minus119890
2119894((1205873)+120587120591) minus1199026119890minus2119894((1205873)+120587120591)
)
times 119891(minus1198902119894((1205873)minus120587120591)
minus1199026119890minus2119894((1205873)minus120587120591)
))minus1
times (1) (119891 (minus1198902119894((1205873)+2120587120591)
minus1199026119890minus2119894((1205873)+2120587120591)
)
times 119891(minus1198902119894((1205873)minus2120587120591)
minus1199026119890minus2119894((1205873)minus2120587120591)
))minus1
(49)
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Numbers
Recall the identity [16 Eq (3 1)]
119891 (minus1198902119894((1205873)minus119911)
minus119902119890minus2119894((1205873)minus119911)
)
times 119891 (minus1198902119894((1205873)+119911)
minus119902119890minus2119894((1205873)+119911)
)
=119890minus2119894119911
(119902 119902)3
infin
119890(minus1198942120587)3(1199023 1199023)infin
119891 (minus1198906119894119911 minus1199023119890minus6119894119911
)
119891 (minus1198902119894119911 minus119902119890minus2119894119911)
(50)
Using the above identity in (49) we obtain the followingEisenstein series identity
Theorem 9 Let |119902| lt 1 Then
infin
sum119899=1
(119899
3)119902119899minus 1199022119899minus 1199024119899+ 1199025119899
1 minus 1199026119899
=1205782(3120591) 120578 (120591) 120578 (9120591) 120578 (18120591)
1205783 (6120591)119866 (119902)
=120578 (120591) 120578 (2120591) 120578 (9120591) 120578 (18120591)
1205782 (6120591)119883 (119902)
(51)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the referees for their severalhelpful comments and suggestions The first author is thank-ful to the University Grants Commission Government ofIndia for the financial support under the Grant F5102SAP-DRS2011 The second author is thankful to DST NewDelhi for awarding INSPIRE Fellowship (no DSTINSPIREFellowship2012122) The third author is thankful to UGCNew Delhi for UGC-BSR fellowship under which this workhas been done
References
[1] S RamanujanNotebooks Vols 1 2 Tata Institute of Fundamen-tal Research Bombay India 1957
[2] S Ramanujan Collected Papers Cambridge University PressCambridge UK 1927 repriented by Chelsea New York NYUSA 1962 reprinted by the American Mathematical SocietyProvidence RI USA 2000
[3] S RamanujanTheLostNotebook andOtherUnpublished PapersSpringer Berlin Germany Narosa Publishing House NewDelhi India 1988
[4] Z-G Liu ldquoA theta function identity and the Eisenstein series onΓ0(5)rdquo Journal of the Ramanujan Mathematical Society vol 22
no 3 pp 283ndash298 2007[5] HH Chan S H Chan and Z-G Liu ldquoTheRogers-Ramanujan
continued fraction and a new Eisenstein series identityrdquo Journalof Number Theory vol 129 no 7 pp 1786ndash1797 2009
[6] C Adiga T Kim M S Mahadeva Naika and H S Madhusud-han ldquoOn Ramanujanrsquos cubic continued fraction and explicit
evaluations of theta-functionsrdquo Indian Journal of Pure andApplied Mathematics vol 35 no 9 pp 1047ndash1062 2004
[7] S Bhargava K R Vasuki and T G Sreeramamurthy ldquoSomeevaluations of Ramanujanrsquos cubic continued fractionrdquo IndianJournal of Pure andAppliedMathematics vol 35 no 8 pp 1003ndash1025 2004
[8] H H Chan ldquoOn Ramanujanrsquos cubic continued fractionrdquo ActaArithmetica vol 73 no 4 pp 343ndash355 1995
[9] K R Vasuki A A Abdulrawf Kahatan andG Sharath ldquoOn theseries expansion of the Ramanujan cubic continued fractionrdquoInternational Journal of Mathematical Combinatorics vol 4 pp84ndash95 2011
[10] K R Vasuki N Bhaskar and G Sharath ldquoOn a continuedfraction of order sixrdquo Annali dellrsquoUniversita di Ferrara vol 56no 1 pp 77ndash89 2010
[11] C Adiga B C Berndt S Bhargava andG NWatson ldquoChapter16 of Ramanujanrsquos second notebook theta functions and q-seriesrdquoMemoirs of the American Mathematical Society vol 315pp 1ndash91 1985
[12] CAdiga andN SA Bulkhali ldquoIdentities for certain products oftheta functions with applications to modular relationsrdquo Journalof Analysis amp Number Theory vol 2 no 1 pp 1ndash15 2014
[13] C F Gauss ldquoHundert Theorems uber die neuen Transscen-dentenrdquo in Werke 3 pp 461ndash469 Konigliche Gesellschaft derWissenschaften zu Gottingen Gottingen Germany 1876
[14] Z-G Liu ldquoA three-term theta function identity and its applica-tionsrdquo Advances in Mathematics vol 195 no 1 pp 1ndash23 2005
[15] S McCullough and L-C Shen ldquoOn the Szego kernel of anannulusrdquo Proceedings of the AmericanMathematical Society vol121 no 4 pp 1111ndash1121 1994
[16] Z-G Liu ldquoA theta function identity and its implicationsrdquoTransactions of the American Mathematical Society vol 357 no2 pp 825ndash835 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of