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NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 165-180
ON MODELS OF ^-REPRESENTATIONS OF sl(2, C) AND (/-CONFLUENT HYPERGEOMETRIC FUNCTIONS
V lV E K SAHAI AND SHALINI SRIVASTAVA
(Received October 2001)
Abstract. We construct new two variable models of irreducible ^-representations of the Lie algebra sl(2, (D). The q-Mellin transformaton is defined and used to transform the models in the form of ^-difference dilation models, involving basic hypergeometric function 2<t>i as basis functions. All the models culminate in many special function identities and recurrence relations.
1 . Introduction
Manocha [6] introduced the idea of irreducible (/-representations of the special complex Lie algebra si (2, C) and developed theory centering around it. A (/-analogue of fractional calculus technique [1, 7] was used to construct models of irreducible (/-representations of si (2, C) followed by the Weisner’s method [18] to obtain identities involving (/-special functions of one and several variables. These models were in terms of (/-differintegral operators acting on (/-hypergeometric functions 20i- Further, in [15], (/-differintegral operator models of sl(2,C) with basis functions involving k2+i(f>k2 functions were constructed and a good number of identities were derived from them.
The theory of ^-representation of Lie algebras together with the theory of q- integral transformations has also been a rich source of results in (/-special function theory. In Sahai [12, 13, 14], the (/-Euler integral transformation, which is motivated by (/-integral representation of beta function, is utilized to obtain models in terms of (/-difference dilation operators of si (2, C) along with identities involving (/-special functions. Both these models, viz. (/-differintegral operator models and (/-difference dilation operator models, are of (/-representations D q (u, a) and (u) of si (2, C) in the particular case u = 0.
In this paper, we construct new models of (/-representations D q (u , a) and f q (u) of si (2, C) corresponding to u ^ 0 acting on (/-confluent hypergeometric functions i<£i. We then define g-Mellin transformation, which is motivated by (/-integral representation of gamma function. The (/-Mellin transformation is the ^-analogue of Mellin integral transformation used in [5, 10]. We utilize this transformation to transform these models of si (2, C) to new models involving (/-difference dilation operators with the (/-hypergeometric functions 20i as basis functions. The discussion culminates in interesting identities and recurrence relations involving i 0 i and 20i functions.
Section-wise treatment is as follows.
2000 A M S Mathematics Subject Classification: 33D80, 17B37.
166 VIVEK SAHAI AND SHALINI SRIVASTAVA
In Section 2, we give definitions involving hypergeometric and basic hypergeometric series, needed for our discussion. In Section 3, Lie algebra si (2, C) is presented along with its g-deformed version. In Section 4, we construct model of D q (—7 / 2, a) and f q (—7 / 2) in two variables acting on i<̂ i and obtain identities and recurrence relations based on it. In Section 5, we introduce g-Mellin transformation and then utilize it to transform the models given in Section 4 to ^-difference dilation models acting on basis functions 2<f>i- The transformed models are further used in obtaining identities and recurrence relations involving 20i- Finally, in Section 6, we give an application of these techniques to g-Laguerre polynomials.
2. Preliminaries
The generalized basic or (/-hypergeometric series r0s is defined as [3]
P*.( ;,,«) = IV r V w S\ O i . . . , b s J A^ ( b 1:q)ri- - - (bs:q)n (q:q)n .n=0(2.1)
where g-shifted factorial (a; q)n is defined by
(2-2)
In other words,( (1 - a) (1 - aq) ■ ■ • (l - aq™-1) , n = 1 , 2 , . . .
(■a'^)n ~ { 1; n==0 (2-3)[ [(1 - a q -1) (l - aq~2) ■ ■ ■ (1 - aq~n)} ; n = - l , - 2 , . . . .
The series r(f>s terminates if one of the numerator parameter is of the form q~m, m = 0 ,1 ,2 ,.. . , and q ^ 0. When 0 < \q\ < 1, the series r0s converges absolutely for all x if r < s; and for |ar| < 1 if r = s + 1. If \q\ > 1 and |jc| < , then alsor<f)s converges absolutely. It diverges for x / 0 if 0 < \q\ < 1 and r > s + 1, and if \q\ > 1 and |a;| > j ~ ~ ^ , unless it terminates.
The generalized hypergeometric series rFs is defined by [9]
\ _ ~ (a i)n ■ ■ ■ (qr)n
r " { A , . . . , A ’ ) ^ 0 (A )n '" ( /3 . ) „ » ! ’ { ' ’
where {a)n is Pochhammer’s symbol defined by
(a) = I a (a + 1) - - ' ( a + n ~ 1) = ^ 5 n = 1 ,2 ,...V ,n \ 1; n = 0. K J
The g-analogue of the binomial function is
. \ = (ffM joo ; |9| < 1; |x | < 1 (2.6)" ) (X'i V oc
The ^-analogues of the exponential functions are
^-REPRESENTATIONS OF sl(2,C) 167
and
Eq 0*0 = Y j Q(n. nV = ’ kl < 1- (2-8)n = 0 ' n
We shall also make use of the function
r , ( « ) = SL^ r ( 1 ~ 9 )1~“ (2-9)eq \Q)
defined for a ^ 0 ,-1 , —2 , . . . . This is a g-analogue of the gamma function and satisfies the functional equation
r , ( a + i ) = ^ r , ( a ) . (2.10)
We need the g-analogue of Kampe de Feriet function [16]
C'E’E1 {(-,m m -*■ v) n % (7 j)m+n n f=1 (5j)m n f=1 ( * ' ) „ m\n\
(2 .11)
in the form [6]
±a -.B\B' ( (a ) : (& );(& ') . x , y ^ :E ;E ' { (c) : (c) ; (c') ’ qr r f
Y ' n/=l ( ^ L - h , 1 ':f=l fe ; Q)m nf=l (fy <?)„ ̂n f=1(e:);g)mnf=: i (e$;g)n (g;g)m(g;g)n
For r = s = 0, we denote the l.h.s. of (2.12) simply by
,A:S;jb' / (a) : (b);(b') \9c:E-,E’ ^ (c) • (e) ; (e; ) ’ x,2/y •
The ^-derivative operator is defined by
a , ( / ( * ) ) = (1_ 9)a <2-13>
where the ^-dilation operator Tx is given by
Tx ( /(* )) = /(«*)• (2-14)
The (/-integral [3] is defined by
° ° ooJ f ( t )dqt = ( 1 - 9 ) £ / ( ? n)9n. (2.15)
168 VIVEK SAHAI AND SHALINI SRIVASTAVA
3. Irreducible q—representations o f sl( 2, C)
The complex Lie algebra sl(2,C) = L { S L ( 2,C )}, the Lie algebra of complex Lie group
SL (2, C) = a b c d
: a, 6, c, d G C, ad — be = 1
consists of all 2 x 2 matrices with trace zero, that is,
2 /sl(2,C) = —e
It has a basis
J + =0 -1 0 0
0 0 -1 0
satisfying the commutation relations
= ± j ± , = 2 J ° .
(3.1)
(3.2)
(3.3)
(3.4)
Let Vq be a complex vector space consisting of g-special functions with a basis {4>x | A G S'} such that the functions {f\ = limq_>i <f>\ \ A € S } form a basis of a vector space, say V. Let A(Vq) be the associative algebra of all linear operators on Vq over the complex field.
A ^-representation of si (2, C) on Vq is a mapping pq : si (2, C) —> A (Vq) satisfying the following conditions:(i) pq (ax + by) = apq (x) + bpq (y )
(ii) There exists a Lie algebra representation p of sl(2) on V such that
lim pq (x) (f)X = p[x)f\ ,q-> 1
for all x, y G sl(2, C) and a, b G C .The representation pq of sl(2, C) is said to be irreducible if there is no proper
subspace W q of Vq which is invariant under pq.Define
J f = P q( J + ), J < = P q(J~), J°q = P q( J 0), (3.5)
where J+, J~, ./“ S A (V q).Manocha [6] has defined the following commutator rules for J^-operators:
J°qJ + - q J + J ° t = J+,
q j p ~ - J~ J°q = - J ~ ,
q J + J - - J ~ J + = 292“ J , ° - ( 1 - 9 ) 92" ^ J “ , u e C . (3.6)
These commutator rules were later generalized by Sahai [11] for the 4-dimensional complex Lie algebra Q (a, b), a, b G C. Further, the commutation relations (3.6) are also equivalent to those introduced by Jimbo [4]. For details, see [6, 14, 15].
If we define an operator Cq on Vq by
c q = q J tJ 7 q ’ q q2UJq (3.7)
^-REPRESENTATIONS OF sl(2,C) 169
then it is easy to check that
(3.8)
As q —> 1, the operators Jq , J~ , Jq reduce to J + , J~, J° which satisfies the commutation relations obeyed by si (2, C) and the operator Cq reduces to the Casimir operator C. Hence, as q —*• 1, pq reduces to a Lie algebra representation p of sl(2,C). For more details, see [6, 8].
4. Models of Irreducible q—representations
We give below new two variable models for each of the ^-representations D q («, a) and (u) corresponding to u = —7 / 2, in which case they will be denoted by D q (—7 / 2, 0;) and ] q (—7 / 2), respectively.
4.1. Representation D q (—7 / 2, a).Consider an irreducible representation D q (—7 / 2, a) of si (2, C), a, 7 G C and 7 ^
0 ,-1 , —2 ,.. . on the representation space Vq with basis {f\ | A = a + n, n G Z}, such that action of si (2, C) on Vq is given by
qx- < h . (4.1)
To find a realization of (4.1) in terms of (/-dilation operators, we choose
j ; = T - 1 ( g - ’ Tt - % - q~< zTzTt) ,
0 _Q 1 „ ’
(4.2)
with basis functions
A = a, a i l , a ± 2, . . . .
170 VIVEK SAHAI AND SHALINI SRIVASTAVA
Model (4.2) obeys (4.1) and the following
J q J q — Q Jq J q = ^ ,
qJ°qJ - - J - J aq = - J ~ ,
q J + J - - J - J + = 2<TV° - (1 - q)q-'lJ°q Jj, 7 S C
as well asqJ^C, = cqj+,
j - c , — qCqJq
JqC, — C T°— Q g '
4.1.1. Identities based on model (4-2).As
fx cZ,t) = 101 ^
satisfiesr _ (1 - < T ^ ) (1 - .
( 1 - 9 )It immediately follows that
. ( . .< ) . * g :
f * (a; «)„(*»';«)„ (6"; « )„ , / \ . a
where ga = a, also satisfies
( 1 - 9 )Using the fact that
Cq (qsJq ) Cq — Cqeq (s ) , which follows from (4.4), we have
c , [ , . K - ) , ]7-1(1-,).9J q>
v /1:0;2 f a : —;b ,b z,qt \ . X ___ • 07-c' > 1 I (^?)
where
^-REPRESENTATIONS OF sl(2,<C) 171
Using the expansionOO
[eq ( s J * )u ] (z , t ) = £ i n f a+n(z, t) , (4.10)n —0
where in are obtained by putting 2 = 0, we get the following identity, after suitable rescaling:
( f f ;gL ,i:Q;2 f a - - , b ' , b " z , - t / WM i l l q, 1
^ (o;«)„ / —,A n a ( 1 qn+~'\ A ( aqn \= 1 j ( ^ ( ? *) 301 ( c' J 1̂ ^ (4'U)
where it; = —l - q
4.1.2. Recurrence relations from model (4-2).The recurrence relations originating from the action of J+, J~ on f\ are given
by
(l - q x) (A + l\z) + qX1(j)1 (\\qz) = i(j>i(\;z), (4.12)
' 1^1 (A;<jz) + 1^1 (\\qz) - (A;z) = (1 - qx~~l) i^i (A - l;gz)(4.13)
respectively, where A (A; *) stands for ^
4.2. Representation |q ( —7 / 2 ).We look for an irreducible representation | 9 (—7 / 2) of sl(2,C),
7 G C — {0, —1, —2 ,... }, on the representation space Vq with basis functions {/a I A = 0 ,1 ,2 ,... } such that the action of si (2, C) on Vq is given by
7 _ nXJ + h = ------— fx+ 1 ,1 - 9
_ 1 — 0̂Jq f x = - ~ j -------------- f X - 1 ,
q 1 - q
T°qfx =q 1 - q
J q f x = ----- 71 ----------- /A ,9 l - q
(yJq Jq + 9 j ? - 9“ % ° ) A = 7/2+1 ̂ 9A/a- (4.14)(1 - 9)
172 VIVEK SAHAI AND SHALINI SRIVASTAVA
To meet this requirement, we choose
J q = y ( < r 7 - J y r , - z 7 y r t )
Jq = — r - ' m - r , ) ,
rO 1 ~ 17/ 2Ttq ~ 1 - 9 ’
C, = q J p ~ +q~'<J0qJ0q - q - ‘J0q,
with basis functions
h = 1̂ 1 ( ;9> -9A+7^ * A> A = 0 ,1 ,2 , . . . .
Model (4.15) satisfies (4.3) as well as (4.4).
4.2.1. Identities based on model (4-15).As
f\ = i0 i ^ qql ; g , - g A+7^ * \
satisfies
C ,A (*, t) = ^ (2i t))( 1 - 9 )
it immediately follows that
» (* -« ) = -I;?;:; )
= E 7 7 ( V' 101 ( <£” ;9 ’ - « " +72) ± ^ ( c ' ; < ? ) n (g;<z)n V 7n=0 v ’ ^/n
satisfies
c qU{z,t) =( 1 - 9 )
Using the fact that
E , ( i SJ “ ) C , = C ,£ , K " ) ,
which follows from (4.4), we have
(1 - <r7/2) (1 - q - r ^ 1)C„ [Eq (sJq ) u] (z, t) =
( 1 - q f
,W l ( (1=5)1 . - q 1+xzt,qt01:1:0 V «2.1
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
, (4.20)
^-REPRESENTATIONS OF sl(2,C) 173
where
[Eq (s J - ) u] (z, t) = ( ° : J f y l ; ) . (4.21)
The expansionOO
[Eq (Sj~ ) U] (z, t) = ^ An fn (z, t) , (4.22)71=0
leads to the following identity
/i:0;i ( a : —; —w /t - q * z t , t \{ d : </7 ; — ’ q\ l )
= S d E f c k 1 0 1 ( s ;< H ^ ( r ( 4 2 3 )
4.2.2. Recurrence relations.The recurrence relations arising from model (4.15) are given by
101 (A;*) - gA+7i0 i (\\qz) - ?A+72i0i (\;qz) = (l - qX+1) i0 i (A + 1;z ) ,(4.24)
i(f)i (\;qz) - qxi(j)i (A;z) = ( l - qx) i0 i (A - 1;q z ) , (4.25)
where i<j>i (A;z) stands for i4>i ( ;q, —qx+7z.
5. Transformed Models of si (2, C)
We introduce the (/-Mellin transformation Iq. Define
13(13-1) 00h (/?, x ) = Iq [f (wz)] = ^ - - y J eq ( - u ) u0 V (ux) dqu, (5.1)
q owhich is motivated from the (/-integral representation of gamma function, as in[17],
OO
r q (7 ) = J eq (-it ) u7_1 dqU. (5.2)0
Using the substitution 2 = ux, which gives z A z — u A u = x A x and Tzf (z) = f (qz) = Tuf (ux), we have the following transforms of certain operator expressions which are needed for the discussion.
Iq [uf (ux)] • = Eph ((3, x ) ,
Iq [uAuf(ux)\ = x A xh (P, x ) , (5.3)
where Eph (/?, x) = h ((5 + 1, x).
174 VIVEK SAHAI AND SHALINI SRIVASTAVA
To obtain transformed models of D q (—7 / 2, 0;) and |q (—7 / 2), we put z = ux in models (4.2) and (4.15). Note that if pq is an irreducible representation of sl(2,C) on the representation space Vq having the basis vectors {f\ | A € S} in terms of { J + , J ~ , J q }, then the transformation Iq induces another irreducible representation crq of sl(2,C) on the representation space W q — IqVq in terms of { i f+ = IqJ + I q l , K q = I g J - I - ' t K ® = Iq j j j l - 1} with basis vectors {hx = I „ h I A € S}.
5.1 . Transformed Model of D q ( —7 / 2, 0:).
K i = r r ^ ( 1 ~ TxTt)'
K~ = ^ T - 1 (q->Tt ~Tx - ^ - ^ E l3q-''xTxTt
K = 1 Zf ^ i . (5-4)1 - q
with basis functions as
h\ [x,t) = 20i ( Q ’ Q~ ;q
Indeed,
and
9 7 ( 1 - 9 ) ? ' 5
K°qK + - q K + K ° q = K + ,
gK°qK q - K q K°q = - K ~ ,
q K + K ~ - K ~ K + = 2q~1K q — (I — q) q~y K qK q (5.5)
K + h x = q->'— ^ h x+l, q 1 ~Q
1 — aA-7/ 2K q h \ = — ^ ---------h > ,
q 1 - Q
Cqh\ = { q K + K q +q -> K °qK 0q -q- '>K°q) h x
- < - - < - 7 ( . . . )(! - q)
Moreover,qK+C'q = C'qK + ,
K~C'q = qC'qK q ,
K qC'q = CqK q. (5.7)
^-REPRESENTATIONS OF sl(2,C) 175
5.1.1. Identities based on transformed model £>g (—7 /2 , a).As
is a solution of
(*, t) = ( ! ~ g 7/2) ( i ~ £ 7/2+1) qx - , hx (X] t) _ (5.8)( 1 - 9 )
the function
w(x,t) = ( aJ qq̂ c>b ^ l) ta
S (c ' ; g )n (g ;g ) n V ? ( i - 9 ) r /
is also a solution of
C > (x ,t ) = 7/2^ 1 ~2g 7/2+1V - 7 t»(*,t)- (5.10)( 1 - 9 )
Using q K + C q — C'qK + , we have
C' [eq [sKq ) u] (z, i) =( ^ f r ^ ; g ) ^ ,-■> (1 - g - ^ ) (1 - r - ' W )
( 1 “ 9 )2
x o;;!;? ( W , (5 .1 1 )V <r»-1(i-9) • 9 ’ 1 ~ ̂ /
where
r_ ( ~ ts+\ -.1 a : q 0]b',b" q ?x /c 1oN[eqr J llj (x, t) ~ f st ̂ ̂ ast . ^7. ^ 1 ̂ ̂ 51J t . (5.12)5 9 L ‘
From the expansionOO
\eq (sKq ) wj (x, i) = ^ ̂A.nha-\.n (x, £),n=0
where An is found by putting x = 0, we arrive at the following identity after suitable rescaling:
( ^ ;9)oo di W a : < f - , V , W \;1 v # : < f - y ’ 1 )
\ q̂ J OO
(l - q)qf> ,(5.13)
176 VIVEK SAHAI AND SHALINI SRIVASTAVA
5.1.2. Recurrence relations from model (5.4)-The recurrence relations originating from the action of K + and K ~ on h\ are
given by
20i (A;/?;x) - qx2(f>i {\-,(3;qx) = (l - qx) 2<t>i (A + 1; /?; x ) , (5.14)
xgA-7- / ? l _ J L 20x (A;/3 + 1 ; gx) + 20i (A; /?; gx) - qx~^20i (A; /?; x)
= (1 - 9A_7) 20i (A - 1; /?; q x ) , (5.15)
( qx qP \where 20i (A; /?; x) stands for 20i ( ; q, \T-̂ )q* ) •
5.2. Transformed Model of tq (—7 / 2).
( « -7 - TxT, - Ed xTxTt\ , (5.16)
k - = ^ r - 1 (*A I - t A t) ,
1 - «7/2T(
with basis functions as
o~x aP n^+i-P \ h\ (x, t) = 201 ( ’ ql ; q, y _ ~ ^ ) 1 ■
Model (5.16) satisfies (5.5) and (5.7) and
Kqh\ =q - T - q \
l - q kx+11
K ’ hx = 1-< -e
'to. »
1 1
1—1 rH
1
1 _ qA +7 /2K°qhx = ^ -------- fcA,1 - g
C'qhx = ( q K + K - + q ~ '
(1 - g-7/2) (1
( 1 - 9qxhx. (5.17)
^-REPRESENTATIONS OF sl(2,C) 177
5 .2 .1 . Identities based on transformed model t q ( —7 / 2).
As/ 0 -A a(3 QX + l - 0 \
h\{x , t ) = 201 f ' ql -,q, x j t , A = 0,1,2,
satisfies
C'qh\ (x,t) = (1 ~ Q 7/2^ 1 ~29 q*hx (x, t) . (5.18)( ! - 9 )
This, in turn, gives
, .x /1:1 ;o ( a : q13; —u ( x , t ) = 0 1:1O , ; 1-9 ’\ c . q ' , - 1
h (c';q)n («;?)„ 2011 ?7 ,9, 1 - 1 ) ’ 1 'n=0 v ’ ’ /n v̂ ’ ^/n
satisfies
C > (x, t) = (1 g 7/2) (1 / " /2+1) 9" « (x, t ) . (5.20)( I - ? )
Using C'q = qCqKq , we have
C'q [E, (s K ~ ) u] (x, t) = (1 g 7/(2l - g / 7/2+1)
(5.2D\ c -0 > 9,1 /
where/ „ • r̂/3- £ j. \
[£ , ( * jr - ) u ] ( x , t ) = J . % q)L ; ’ * ) - (5-22)\ c • y » q->1 /The expansion
OO[E9 w] (x, t) = ^ 2 Cnhn {x, t) , (5.23)
n=0gives rise to the following identity:
. ~/3. it; q~ pxt •a -q t .
r ' • rH • — ’ ic - q \ ~ q, 1
(a;g)„ m ( aQn ^ jl / 9“ n,9/3 9n+7 p V= t a r o s : 1011 o v ^ tn=0
(5.24)
178 VIVEK SAHAI AND SHALINI SRIVASTAVA
5.2.2. Recurrence relations.The recurrence relations that come from (5.17) are given by
Q
= ( 1 - < 2 A) 2< M A + 1;/?;x), (5.25)
20i (A; 0 ; x ) - gA+720i (A; (3;qx) - xqx+1 13 ^ _ Q 20i (A; (3 + 1; qx)
20i (A;/3;qx) - qx2(j>i (A; /5; a?) = (l - qx) 20i (A - 1; /?; qx) , (5.26)
( q~x q^ a+7—/3 \’ ^7 ;</, q--[_q x J .
6. Conclusion
In this section, we give an application of our models to g-Laguerre polynomials. Let a > — 1 and 0 < q < 1 . Then g-Laguerre polynomials are defined by [3]
ra ( \ _ + )n , f q ” _ n+a+ A . s9/ (qm q) ̂ \ y ' 6.1 )
If we take 2: = (1 — q) x then it can be easily seen that l im ^ i- (z; q) = (x), where (x) are classical Laguerre polynomials. Consider the model (4.15) in the particular case 7 = 1 + a (that is, a model of representation ] q (— (1 + a) / 2)) with basis functions involving g-Laguerre polynomials, given as
4 = (■T(I+“ ) - TzTt - zTzTt) ,
-1
_ 1 - g(l+»)/2Tt
5 1 - 9
/ „ (* ,* ) = L ° ( z ; q ) t n. (6.2)The action of these Jg-operators on f n (z, t) will take the following form:
7n+lj f f n = r ( 1 + a ) \ * q f n + 1 ,
J— f __ ^ ru a Jn — J n —1>1 - q
14-a1 /Tf 2 ”1^= - ..f n - (6.3)
We can obtain identities involving ^-Laguerre polynomials using the method explained above. For example, taking
(*’ () = § J r i A 101 ( ; 9’ ~29"+a+1) <n ( 6 '4 )n=0
^-REPRESENTATIONS OF sl(2,C) 179
and proceeding as in Section 4.2.1, we arrive at the following identity:
00 fn
= Ln (^<1) 1^n= 0 ' ,q)n
(6.5)
which is a special case of (4.23).The recurrence relations involving g-Laguerre polynomials can also be obtained
by using first two equations of (6.3). These will shape as
L “ ( z ; q ) - q 1+ °‘+nL Z ( q z ; q ) - z q n+a+1L°(qz ;q) = (l - ,"+>) L«+1 (z; 9) (6.6)
respectively. Eliminating (z; q) from (6.6) and (6.7) gives rise to a three term recurrence relation as follows
(l - qa+n) £ “ _ ! (qz:q) + (zq1+a+2" - ( l - <j1+“ +2" ) ) £ “ (qz;q)
+ qn ( l - q n+1) K + l (z-,q) = 0, (6.8)
Note that (6.8) is a g-analogue of the three term recurrence relation for classical Laguerre polynomials [2],
(.a + n) (x) 4- (x — 1 - a - 2n) L% (x) + (n + 1) L“ +1 (x) = 0. (6.9)
Acknowledgem ents. The authors thank the referee for his valuable suggestions which led to a better presentation of the paper.
1. M.A. Al-Bassam and H.L. Manocha, Lie theory of solutions of certain differ- integral operators, SIAM J. Math. Anal. 18 (1987), 1087-1100.
2. G.E. Andrews, R.Askey and R. Roy, Special Functions, Cambridge Univ. Press,
3. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, 1990.
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and
qnL Z ( z ; q ) - L Z { q z ; q ) = - (l - qa+n) L ^ (qz; q) (6.7)
n = 1,2, . . . .
References
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180 VIVEK SAHAI AND SHALINI SRIVASTAVA
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Vivek SahaiDepartment of Mathematics and Astronomy Lucknow University Lucknow 226007 INDIA
Shalini SrivastavaDepartment of Mathematics and Astronomy Lucknow University Lucknow 226007 INDIA