kadell k w j - a simple proof of an aomoto-type extension of askey's last conjectured selberg...

8/9/2019 Kadell K W J - A Simple Proof of an Aomoto-Type Extension of Askey's Last Conjectured Selberg Q-Integral - J. Math.

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Journal of Mathematical Analysis and Applications 261, 419440 (2001)

doi:10.1006/jmaa.2000.7180, available online at http://www.idealibrary.com on

A Simple Proof of an Aomoto-Type Extension of

Askeys Last Conjectured Selberg q-Integral

Kevin W. J. Kadell

Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804

Submitted by Bruce C. Berndt

Received June 1, 1998

We establish an Aomoto-type extension of Askeys last conjectured Selbergq-integral, which was recently proved by Evans. We follow the lines of our proofsof Aomoto-type extensions of the Morris constant term q-identity and Gustafsons

AskeyWilson Selberg q-integral. We require integral forms of the q-transportationtheory and its alternative for the root system An1, which are related to the simplereflections and minuscule weight of An1. We use an elementary symmetry whichis related to the minuscule weight of An1 to lift a proof of the one dimensionalq-integral to the multivariable setting. 2001 Academic Press

Key Words: Selbergs integral; Aomoto-type extension; q-transportation theory forthe root system An1.

1. INTRODUCTION AND SUMMARY

Throughout this paper, we let n 1, k 0, N 0, a 0, b 0, m, andv be integers with 0 m n and 2 v n, we let x and y be complex

with positive real parts, and we let q be real with 0 < q < 1. In 1944,Selberg [27] evaluated the multivariable beta integral

1

0

1

0

ni=1

tx1i 1 tiy12kn t1 t ndt1 dtn

=n

i=1

x + n iky+ n ik1 + ik

x + y+ 2n i 1k 1 + k (1.1)

where nt1 t n =

1i


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420 kevin w. j. kadell

q-beta integrals given by Ramanujan [25, 26]; see also Askey [4] andHardy [13]. Gustafson [10, 11] has proven many extensions of Selbergsintegral including extensions based upon MellinBarnes type integrals and

the AskeyWilson integral with five parameters.Let z qa =

a1i=0 1 q

iz be the q-Pockhammer symbol. FollowingAskey [5], we set

q2kn t1 t n =

1i


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askeys last selberg q-integral 421

by the asymptotic relation

qxn2k

qIkn x n 1k + 0 C1 q

n (1.9)

where the constant C is independent ofx and y. Kadells proof generalizedAomotos proof [3] of an extension of Selbergs integral. See Evans [7] fora proof using Andersons argument [1].

Let w f denote the coefficient of the monomial w in the Laurent expan-sion of f. Habsieger [12] and Kadell [17] independently observed that theSelberg q-integral (1.7) is equivalent to the Morris constant term q-identity

1n

i=1

ti qaq/ti qb qakn1t1 t n

=n

i=1

q qa+b+nikq qik

q qa+nikq qb+nikq qk(1.10)

A natural extension of the Jackson q-integral (1.5) is given by

dc

ftdqt = c

0ftdqt+

d0

ftdqt

= c1 q

i=0

qifqic + d1 q

i=0

qifqid (1.11)

which is the Riemann sum using the endpoints which are furthest from zero

in the partition

c dq = qic i 0 qid i 0 (1.12)

Andrews and Askey [2] have given the q-beta integral

qIx y c d =

d

c

qt/c qqt/d qqxt/c qq

yt/d qdqt

=cd

c + d

d/c qc/d qqxd/c qq

yc/d q

qxqy

qx + y (1.13)

Observe that the integrand has no poles with t c dq.Omitting m as a subscript when m = 0, we set

qaskn mx y c d t1 t n

=n

i=1

qti/c qqti/d qqx+ni+1mti/c qq

yti/d qq

2kn t1 t n (1.14)


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where A is one or zero according to whether A is true or false, respec-tively, and we use capital letters to denote the q-integral

qASk

n mx y c d

=d

c

dc

qaskn mx y c d t1 t ndqt1 dqtn (1.15)

Askeys last conjectured Selberg q-integral [5], which has been provenby Evans [8], reduces to (1.13) when n = 1. This is the m = 0 case of thefollowing theorem, which is our main result.

Theorem 1.

qASkn mx y c d = q

n3k2 n2

k2cd

n2k

n

i=1

cdd/c qc/d qc + dqx+nik+imd/c q q

y+nikc/d q

n

i=1

qx + n ik + i mqy+ n ikq1 + ik

q

x+

y+ 2

n

i 1

k+

i

m

q1 +

k

(1.16)

Observe that the integrand has no poles with t1 t n c dnq, that the

integrand is bounded on c dnq, and that the case cd = 0 of (1.16) is Askeysfirst conjecture (1.7).

Krattenthaler [22] used the case m = 0, k = 1 of Theorem 1 toobtain exact enumeration formulas for perfect matchings of holey Aztecrectangles.

Observe by (1.2) and (1.14) that we have the symmetry

qaskn x y c d t1 t n = qas

kn y x d c t1 tn (1.17)

and that dc

d

cft1 t ndqt1 dqtn

=

d

c

d

cf t1 tndqt1 dqtn (1.18)

Using (1.15), (1.17), and (1.18), we have the symmetry

qASkn x y c d = qAS

kn y x d c (1.19)

We follow the lines of our proofs [19, 20] of Aomoto-type extensions ofthe Morris constant term q-identity (1.10) and Gustafsons AskeyWilsonSelberg q-integral. We require integral forms of the q-transportation the-ory for the root system An1 and its alternative. These are related to the


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properties of the simple reflections and the minuscule weight of An1. Weuse the elementary symmetry

qakn1qtn t1 t n1 = qa

kn1t1 t n (1.20)

See Carter [6], Grove and Benson [9], and Kadell [1820] for details on theproperties of root systems. The q-transportation theory for root systems hasa long history in [1620] and, in various forms, in many of the papers onconstant term q-identities associated with root systems.

In Section 2, we use the substitution t = qs to give a simple relation and,using a boundary condition, we prove the AndrewsAskey q-beta integral

(1.13) when x is a positive integer.In Section 3, we give an integral form of the q-transportation theory for

the root system An1 which we express in terms ofqaskn x y c d t1 t n.

In Section 4, we lift the simple relation of Section 2 to the multivari-able setting, obtaining an integral form of the alternative q-transportationtheory for the root system An1 which we express in terms of qas

kn x y

c d t1 t n.

In Section 5, we establish the dependence of qAS

k

n m x y c d on theparameters m and x.In Section 6, we recall the global form [16, Sect. 4] of the q-transportation

theory for the root system An1 and give an application.In Section 7, we follow Askey [4] and lift the boundary condition of Sec-

tion 2 to the multivariable setting, obtaining a recurrence relation involvingthe parameters x, c, and n.

In Section 8, we use induction on n, x, and m to evaluate qASkn m

x y c d when x is a positive integer and, using Ismails argument [15],we extend to complex x and complete the proof of Theorem 1.

2. A PROOF OF THE ANDREWSASKEYq-BETA INTEGRAL (1.13)

In this section, we use the substitution t = qs to give a simple relationand, using a boundary condition, we prove the AndrewsAskey q-beta inte-gral (1.13) when x is a positive integer.

Observe that the q-differential

dqft = ft fqt (2.1)

satisfies the scale invariance property

dqct

ct=

dqt

t= 1 q (2.2)


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The q-derivative is given by the ratio of q-differentials

dq

dqtft = dqft

dqt

=1

t1 qft fqt (2.3)

We define

qN

ftdqt = 1 qN1i=0

qifqi (2.4)

to be the finite sum which results when we cancel the possibly divergent

infinite series in the definition (1.11). We have the principal value of theq-integral

PV

dc

ftdqt

= lim

N

qNcc

ftdqtqNd

dftdqt

= limN

c1 qN1

i=0qifqic

+ d1 qN1i=0

qifqid (2.5)

which extends the definition (1.11).The following lemma gives an integral form of the alternative q-

transportation theory for the root system An1.

Lemma 2.If gt has no poles with t c dq, then we have

PV

dc

1 + t/c1 t/dqt/c qqt/d q

qxt/c qqyt/d q

gtdqt

t

= PV

dc

1 + qxt/c1 qyt/dqt/c qqt/d q

qxt/c qqyt/d q

gqt

dqt

t

(2.6)

Proof. Observe that

ft = 1 + t/c1 t/dqt/c qqt/d q

qxt/c qqyt/d q

gt (2.7)

has no poles with t c dq and

fqt = 1 + qxt/c1 qyt/dqt/c qqt/d q

qxt/c qqyt/d q

gqt (2.8)


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Hence we may write the result (2.6) as

PV d

cft

dqt

t = PV d

cfqt

dqt

t (2.9)Observe by (2.7) that

0 = f c = fd (2.10)

Replacing ft by ft/t in (2.5), we obtain

PV

dc

ftdqt

t

= 1 q lim

N

N1i=0

fqic + fqid (2.11)

Using (2.10) and observing that we may shift both of the sums on the rightside of (2.11), we have

PV

dc

ftdqt

t

= PV

qdqc

ftdqt

t

(2.12)

Substituting (2.12) into (2.9), we see that the result (2.6) becomes

PV

qd

qcft

dqt

t

= PV

d

cfqt

dqt

t

(2.13)

which follows using the substitution t = qs, the scale invariance (2.2) ofdqt/t, and replacing s by t.

Observe that1 + t/c1 t/d 1 + qxt/c1 qyt/d

qxdt

= 1 + t1/c 1/d t2/cd 1 tqx/c qy/d + qx+yt2/cd

qxd

t

= qxd/c qx qxt/c q2xd/c + qx+y + q2x+yt/c

= 1 qx + qxd/c q2xd/c 1 + qx+y qxt/c + q2x+yt/c

= 1 qx1 + qxd/c 1 qx+y1 + qxt/c (2.14)

Setting gt = qxd in Lemma 2 (2.6), moving both q-integrals to thesame side of the equation, using (2.14), and observing that the principal

value extends the definition (1.11), we obtain

0 = PV

dc

1 qx1 + qxd/c 1 qx+y1 + qxt/c

qt/c q qt/d qqxt/c qq

yt/d q

dqt

t

= d

c 1 qx1 + qxd/c 1 qx+y1 + qxt/c

qt/c q qt/d qqxt/c qq

yt/d qdqt (2.15)


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which we may rearrange as

1 qx1 + qxd/cqIx y c d = 1 qx+yqIx + 1 y c d (2.16)

Using the functional equation

qx + 1 =1 qx

1 qqx (2.17)

for the q-gamma function, we see that the product on the far right sideof (1.13) satisfies (2.16).

We require the following q-analogue of [16, Lemma 3].

Lemma 3. If gxt is bounded on c dq and gxc is continuous at

x = 0 from the right, then we have

limx0+

1 qx

1 q

dc


yt/d qgxtdqt

=cd

c + d

c/d qqyc/d q

g0c (2.18)

Proof. Using the definition (1.11) and cancelling the factor 1 q, we

obtain

limx0+

1 qx

1 q

dc


yt/d qgxtdqt

= limx0+

1 qx

c

i=0

qiqi+1 qq

i+1c/d qqi+x qq

i+xc/d qgxq

ic

+ d

i=0

qiqi+1d/c qq

i+1 q

qi+xd/c qqi+x qgxq

id= c

qc/d qqyc/d q

g0c (2.19)

which readily simplifies to the result (2.18).

Taking gxt = 1 in Lemma 3 (2.18), we see that qIx y c d satisfiesthe boundary condition

limx0+

1 qx1 q q

Ix y c d = cdc + d

c/d qqyc/d q

(2.20)

Using (1.4) and (2.17), we have

limx0+

1 qx

1 qqx = lim

x0+qx + 1 = q1 = 1 (2.21)

Hence that the product on the far right side of (1.13) also satisfies theboundary condition (2.20).

Combining our results (2.16) and (2.20), we may establish the AndrewsAskey q-beta integral (1.13) by induction on x.


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3. THE q-TRANSPORTATION THEORY FORTHE ROOT SYSTEM An1

In this section, we give an integral form of the q-transportation theory forthe root system An1 which we express in terms ofqas

kn x y c d t1 t n.

Using the identities

z qa = zaq

a2q1a/z qa (3.1)

z qa+b = z qaqaz qb (3.2)

for reversing and splitting the q-Pockhammer symbol, respectively, and thefact that k1 k +

k2

=

k2

, we have

s2kq1kt/s q2k = s2kq1kt/s qkqt/s qk

= s2k q1kt/skqk2s/t qkqt/s qk

= stkqk2s/t qkqt/s qk (3.3)

The following lemma gives an integral form of the q-transportation theoryfor the root system An1.

Lemma 4. If F is a distribution function on the domain and Ys t issymmetric,

Ys t = Yt s (3.4)

in s and t, then we have 2

ts ts QtYs tdFsdFt

= Q

2ss ts QtYs tdFsdFt (3.5)

Proof. Since Qs ts ts Qt is antisymmetric under the substi-tution s t, we have

0 =

2Qs ts ts QtYs tdFsdFt (3.6)

The result (3.5) follows by expanding the factor Qs t in (3.6) and rear-ranging the result.

The following lemma expresses Lemma 4 (3.5) in terms of qaskn x y c d

t1 t n.


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Lemma 5. If t1 t n is symmetric,

t1 t n = t1 t v2 tv tv1 tv+1 t n (3.7)

in tv1 and tv, then we have

PV

dc

d

ctvt1 t n qas

kn x y c d t1 t ndqt1 dqtn

= qk PV

dc

d

ctv1t1 t nqas

kn

x y c d t1 t ndqt1 dqtn

(3.8)

PV

dc

d

c

1

tv1t1 t n qas

kn x y c d t1 t ndqt1 dqtn

= qk PVd

c

d

c

1

tv

t1 t n qaskn

x y c d t1 t n dqt1 dqtn

(3.9)

Proof. Define q2kn v t1 t n by

q2kn t1 t n = tv1 tvtv1 q

ktv q2kn v t1 t n (3.10)

Using (3.3) and the fact that st1 s/t1 qk

t/s = s ts qk

t, wehave

s2kq1kt/s q2k = s ts qkt stk1

qk2qs/t qk1qt/s qk1 (3.11)

Using (3.11) with s = tv1 and t = tv, we see by (1.2) and (3.10) that

q2kn v t1 t n = tv1tvk1qk2qtv1/tv qk1qtv/tv1 qk1

v2i=1

t2ki q1ktv1/ti q2kt

2ki q

1ktv/ti q2k

n

j=v+1t2kv1q

1ktj/tv1 q2k t2kv q

1ktj/tv q2k

1i


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Using (1.14), (3.7), (3.10), and (3.12), we obtain

t1 t n qaskn x y c d t1 t n

= tv1 tvtv1 qktv Ytv1 tv (3.13)

where

Ytv1 tv = t1 t nn

i=1

qti/c qqxti/c q

qti/d q

qyti/d q

q2kn v t1 t n (3.14)

is symmetric in tv1 and tv. The result Lemma 5 (3.8) now follows by apply-ing Lemma 4 (3.5) with Ytv1 tv given by (3.14) and s = tv1, t = tv,Q = qk.

The result Lemma 5 (3.9) now follows by incorporating 1/tv1tv intot1 t n.

4. THE ALTERNATIVE Q-TRANSPORTATION THEORYFOR THE ROOT SYSTEM An1

In this section, we lift the simple relation Lemma 2 (2.6) to themultivariable setting, obtaining an integral form of the alternativeq-transportation theory for the root system An1 which we express interms of qas

kn x y c d t1 t n.

Setting s = ti and t = tj for 1 i < j n in (3.3), we see by (1.2) that

q2kn t1 t n =

1i


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Lemma 6. If gt1 t n has no poles on c dnq, then we have

PVd

c

d

c

1 + t1/c1 t1/d qaskn x y c d t1 t n

gt1 t ndqt1

t1dqt2 dqtn

= qn1k PV

dc

d

c1 + qxtn/c1 q

ytn/dqaskn

x y c d t1 t ngqtn t1 t n1 dqt1 dqtn1dqtn

tn (4.3)

Proof. Observe that

ft1 t n = 1 + t1/c1 t1/dn

i=1

qti/c qqxti/c q

qti/d qqyti/d q

q2kn t1 t n gt1 t n (4.4)

has no poles with t1 t n c dn

q and

fqtnt1tn1 = 1+qxtn/c1q

ytn/dn

i=1

qti/cqqxti/cq

qti/dqqyti/dq

q2kn qtnt1tn1gqtnt1tn1 (4.5)

Using (1.14), (4.2), (4.4), and (4.5), we see that we may write the result (4.3)as

PV

dc

d

cft1 t n

dqt1

t1dqt2 dqtn

= PV

dc

d

cfqtn t1 t n1 dqt1 dqtn1

dqtn

tn

(4.6)

Observe by (4.3) that

0 = fc t2 t n = fd t2 t n (4.7)

where t2 t n c dn1q . Using (4.7) and observing that we may shift

both of the sums on the right side of (2.11), we have

PV

d

c

d

cft1 t n

dqt1

t1dqt2 dqtn

= PV

qdqc

dc

d

cft1 t n

dqt1

t1dqt2 dqtn

(4.8)


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Substituting (4.8) into (4.6), we see that the result (4.3) becomes

PVqd

qc

d

c

d

c

ft1 t ndqt1

t1dqt2 dqtn

= PV

dc

d

cfqtn t1 t n1 dqt1 dqtn1

dqtn

tn

(4.9)

which follows using the substitution t1 t n = qsn s1 sn1, thescale invariance (2.2) of dqt1/t1, and replacing si by ti, 1 i n.

5. THE DEPENDENCE OF qASkn mx y c d ON THE

PARAMETERS m AND x

In this section, we establish the dependence of qASkn mx y c d on the

parameters m and x. Throughout this section, we let 1 m n.We set

= PVd

c

dc

1 + t1/c1 t1/d

ni=nm+2

1 + qxti/c

qaskn x y c d t1 t n

dqt1

t1dqt2 dqtn

(5.1)

Taking gt1 t n =n

i=nm+21 + qxti/c in Lemma 6 (4.3), we obtain

= qn1k

PVd

c d

c1 qy

tn/d

ni=nm+1

1 + qx

ti/c

qaskn x y c d t1 t ndqt1 dqtn1

dqtn

tn

(5.2)

Applying Lemma 5 (3.9) m 1 times to the term 1/tn in

1 qytn

/d1

tn= 1/t

n

qy/d (5.3)

and the q-integral on the right side of (5.2) with v running from n 1to n m + 1 and t1 t n =

ni=nm+11 + q

xti/c and rearrangingt1 t n, we obtain

= PV

dc

d

cqnmk/tnm+1 q

y+n1k/d1 + qxtnm+1/c

n

i=nm+2

1 + qxti/c qaskn x y c d t1 t n dqt1 dqtn

(5.4)


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Applying Lemma 5 (3.8) and (3.9) n m times to the terms t1/cd and1/t1, respectively, in

1 + t1/c1 t1/d 1t1= 1/t1 + 1/c 1/d t1/cd (5.5)

and the q-integral on the right side of (5.1) with v running from twoto n m + 1 and t1 t n =

ni=nm+21 + q

xti/c, and factoring theintegrand, we obtain

= PVd

c

d

c qnmk/tnm+1 + 1/c 1/d qnmktnm+1/cd

ni=nm+2

1 + qxti/c qaskn x y c d t1 t n dqt1 dqtn

(5.6)

Substituting

xycdt x + n mk y+ n 1k

q

nmk

c q

nmk

d tnm+1 (5.7)into (2.14) and rearranging the left side, we obtain

qnmk/tnm+1 + 1/c1 qnmktnm+1/d

1 + qxtnm+1/cqnmk/tnm+1 q

y+n1k/d

qx+nmkd

= 1 q

x+nmk

1 + q

x+nmk

d/c 1 qx+y+2nm1k1 + qxtnm+1/c (5.8)

Equating (5.4) and (5.6), multiplying by qx+nmkd, factoring theintegrand of (5.6), moving both q-integrals to the same side of theequation, using (5.8), and observing that the principal value extendsthe definition (1.11), we obtain

0 = 1 qx+nmk1 + qx+nmkd/c

d

c

d

c

n

i=nm+2

1 + qxti/c qaskn x y c d t1 t ndqt1 dqtn

1 qx+y+2nm1k

d

c

d

c1 + qxtnm+1/c

n

i=nm+2

1 + qxti/c Qaskn x y c d t1 t ndqt1 dqtn (5.9)


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Using (1.14) and (1.15), we may rearrange (5.9) as

qASkn mx y c d = 1 + q

x+nmkd/c

1 qx+nmk

1 qx+2nm1kqAS

kn m1x y c d (5.10)

which gives the dependence ofqASkn mx y c d on the parameter m.

Using (5.10), we see by induction on m that

qASkn mx y c d =

m

i=1

1 + qx+nikd

1 qx+nik

1 qx+2ni1k qASkn x y c d (5.11)

Setting m = n in (5.11) and using the fact that

qASknnx y c d = qAS

kn x + 1 y c d (5.12)

we obtain

qASkn x + 1 y c d =

ni=1

1 + qx+nikd

1 qx+nik

1 qx+2ni1kqAS

kn x y c d (5.13)

which gives the dependence ofqASkn x y c d on the parameter x.

6. THE GLOBAL FORM OF THE q-TRANSPORTATION THEORYFOR THE ROOT SYSTEM An1

In this section, we recall the global form [16, Sect. 4] of the q-transportation theory for the root system An1 and give an application.

Throughout this section, we follow the notation and recall results of[16, Sect. 4].

We set

Qijnt1 t n =

1i


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The case m = 0 of [16, Lemma 4 (4.4)] is

nQi j

nt1 t n = Sn

1i


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which is symmetric,

qhknx y c d t1 t n

= qhknx y c d t1 t n Sn (6.11)

in t1 t n, and we use capital letters to denote the q-integral

qHkn x y c d

=d

c

dc

qhknx y c d t1 t ndqt1 dqtn (6.12)

Setting Qij = Q, 1 i < j n, in (6.3) and using (6.9), we obtain

n

1i


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and, using the antisymmetry (6.4),

q2k1n t1 t n1 c =

n1

i=1

t2k1i q1kc/ti q2k1 q

2k1n1 t1 t n1

= c2k1n1n1i=1

q1kti/c q2k1

q2k1n1 t1 t n1 (7.4)

Observe that

n1i=1

1 + ti/cqti/c q

qxti/c q

x=0

= 1 (7.5)

Substituting (7.3) and (7.4) into (7.2), setting x = 0 and tn = c, andusing (7.5) and the fact that q2k1 = q/q

2k1 q, we obtain

g0c = c2n1k

d

c

d

c

n1

i=1q1kti/c qqti/d q qkti/c qq

yti/d q

n1t1 t n1q2k1n1 t1 t n1dqt1 dqtn1 (7.6)

Since the integrand on the right side of (7.6) vanishes for ti = qjc where

1 i n 1 and 0 j k 1, we may write (7.6) as

g0c = c2kn1

d

qkc

d

qkc

n1

i=1q1kti/c qqti/d qqkti/c qq

yti/d q

n1t1 t n1q2k1n1 t1 t n1dqt1 dqtn1

= c2kn1qHkn12k y q

kc d (7.7)

Since gx satisfies the hypotheses of Lemma 3, we see using (7.1) and (7.7)that Lemma 3 (2.18) gives

limx0

+

1 qx

1 qqH

kn x y c d

= ncd

c + d

c/d qqyc/d q

c2kn1qHkn12k y q

kc d (7.8)

which is [16, (9.17)]. Using (6.14), we see that (7.8) becomes the recurrencerelation

lim

x0+

1 qx

1 q

qASkn x y c d =

1 qnk

1 qk

cd

c + d

c/d q

qy

c/d q c2n1kqAS

kn12k y q

kc d (7.9)


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8. A PROOF OF THEOREM 1

In this section, we use induction on n, x, and m to evaluate qASkn mx y

c d when x is a positive integer and, using Ismails argument [15], weextend to complex x and complete the proof of Theorem 1.

Using the fact that cdd/c q/c + d = dqd/c q, we may writethe function on the right side of (1.16) as

qRkn mx y c d = q

kn c dq

kn mx y c dq

kn mx y (8.1)

where

qkn c d = q

n3k2 n2

k2cd

n2k (8.2)

qkn mx y c d

= dnn

i=1

qd/c qc/d qqx+nik+imd/c qq

y+nikc/d q (8.3)

qkn mx y

=n

i=1

qx + n ik + i mqy+ n ikq1 + ik

qx + y+ 2n i 1k + i mq1 + k (8.4)

Using (2.17), we see that qRkn mx y c d satisfies (5.10) and (5.12).

Observe that

qkn1q

kc d = qn1

3 k2 n12

k2qkcd

n12 k

= qn3k

2 n12 k2cd

n12 k (8.5)

Comparing (8.2) and (8.5), we have

qkn c d = q

n1k2cdn1kqkn1q

kc d (8.6)

Observe that (8.3) gives

qkn 0 y c d = d

nn

i=1

qd/c qc/d qqnikd/c qq

y+nikc/d q (8.7)

qkn12k y q

kc d

= dn1n1i=1

q1kd/c qqkc/d qqnikd/c qq

y+nikc/d q (8.8)


20/22


Using (3.1) and (3.2), we obtain

qd/c qc/d q

q1k

d/c qqk

c/d q

=c/d qk

q1k

d/c qk

= qk2c/dk (8.9)

Using (8.9) and the fact that dqd/c q/d/c q = cd/c + d tocompare (8.7) and (8.8), we obtain

qkn 0 y c d = q

n1k2c/dn1kcd

c + d

c/d qqyc/d q

qkn12k y q

kc d (8.10)

Using (2.17) and (2.21), we obtain

limx0+

1qx

1q qkn xy

=n1i=1

qnikn

i=1

qy+nikq1+ik

qy+2ni1kq1+k

= q1+nkq1+k

n1i=1

qnikqy+n1ikq1+ikqy+2ni1kq1+k

=1qnk

1qk

n1i=1

qn+1ikqy+n1ikq1+ik

qy+2ni1kq1+k (8.11)

qk

n12ky =

n1i=1

qn+1ikqy+n1ik

qy+2ni1k

q1+ik

q1+k (8.12)

Comparing (8.11) and (8.12), we have

limx0+

1 qx

1 q qkn x y =

1 qnk

1 qk qkn12k y (8.13)

Multiplying (8.6), (8.10), and (8.13), we see that qRknx y c d satisfies the

recurrence relation (7.9).Using (5.10), (5.12), and (7.9), and proceeding by induction on n, x, and

m, we may evaluate qASkn x y c d when x is a positive integer.

Recall the identity theorem by Hille [14, Sect. 8.1] that two functionswhich are analytic in the domain and agree at infinitely many points of which include an accumulation point in agree throughout .

Let w and z be complex numbers. Observe that wz q is an entirefunction of z and that if w 1 then 1/wz q is an analytic function of

z in the unit disc around zero. Observe that

s2kq1kt/s q2k = s q1kt s qkt (8.14)


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Setting s = ti and t = tj for 1 i < j n in (8.14), we see by (1.2) that

q2kn t1 t n is a polynomial. Since finite sums and products of analytic

functions are analytic, we see that qASkn mx y c d and qR

kn mx y c d

are analytic functions of z = qx in the unit disc around zero. Since theyhave the same values at z = qx where x is a positive integer and theyare both analytic at z = 0, we see by the identity theorem that they areequal for z in the unit disc around zero. Since q < 1, this is the halfplane where the real part of x is positive. This completes the evaluation of

qASkn mx y c d and establishes Theorem 1.

Our argument was introduced by Ismail [15] who gave a simple proof

of Ramanujans 11 summation formula. Kaneko [21] used Ismails argu-ment to give a multivariable 11 summation theorem using the Macdonaldpolynomials.

REFERENCES

1. G. Anderson, A short proof of Selbergs generalized beta formula, Forum Math. 3 (1991),

415417.2. G. E. Andrews and R. A. Askey, Another q-extension of the beta function, Proc. Amer.

Math. Soc. 81 (1981), 97101.3. K. Aomoto, Jacobi polynomials associated with Selbergs integral, SIAM J. Math. Anal. 18

(1987), 545549.4. R. A. Askey, Ramanujans extensions of the gamma and beta functions, Amer. Math.

Monthly 87 (1980), 346359.5. R. A. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews,

SIAM J. Math. Anal. 11 (1980), 938951.

6. R. W. Carter, Simple Groups of Lie Type, Wiley-Interscience, London/New York, 1972.7. R. Evans, Multidimensional q-beta integrals, SIAM J. Math. Anal. 23 (1992), 758765.8. R. Evans, Multidimensional beta and gamma integrals, Contemp. Math. 166 (1994),

341357.9. L. C. Grove and C. T. Benson, Finite Reflection Groups, 2nd ed., Springer-Verlag,

New York, 1985.10. R. A. Gustafson, A generalization of Selbergs beta integral, Bull. Amer. Math. Soc. 22

(1990), 97105.11. R. A. Gustafson, Some q-beta and MellinBarnes integrals with many parameters associ-

ated to the classical groups, SIAM J. Math. Anal. 23 (1992), 525551.12. L. Habsieger, Une q-integrale de SelbergAskey, SIAM J. Math. Anal. 19 (1988),

14751489.13. G. H. Hardy, Ramanujan, Cambridge Univ. Press, Cambridge, UK, 1940; reprinted by

Chelsea, New York, 1959.14. E. Hille, Analytic Function Theory, Vol. 1, 2nd ed., Chelsea, New York, 1973.15. M. E.-H. Ismail, A simple proof of Ramanujans 11 summation formula, Proc. Amer.

Math. Soc. 63 (1977), 185186.16. K. W. J. Kadell, A proof of some q-analogues of Selbergs integral for k = 1, SIAM

J. Math. Anal. 19 (1988), 944968.17. K. W. J. Kadell, A proof of Askeys conjectured q-analogue of Selbergs integral and a

conjecture of Morris SIAM J. Math. Anal. 19 (1988), 969986.


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18. K. W. J. Kadell, A proof of the q-MacdonaldMorris conjecture for BCn, AMS Memoir108, No. 516 (1994).

19. K. W. J. Kadell, A simple proof of an Aomoto-type extension of the q-Morris theorem,Contemp. Math. 166 (1994), 167181.

20. K. W. J. Kadell, A simple proof of an Aomoto-type extension of Gustafsons AskeyWilsonSelberg q-integral, Methods Appl. Anal. 5 (1998), 125142.

21. J. Kaneko, A 11 summation theorem for Macdonald polynomials, Ramanujan J. 2 (1998),379386.

22. C. Krattenthaler, Schur function identities and the number of perfect matchings of holeyAztec rectangles, Contemp. Math. 254 (2000), 335349.

23. I. G. Macdonald, The Poincare series of a Coxeter group, Math. Ann. 199 (1972), 161174.24. P. A. Mac Mahon, Two applications of general theorems in combinatory analysis: (1) to

the theory of inversions of permutations; (2) to the ascertainment of the numbers of termsin the development of a determinant which has amongst its elements an arbitrary numberof zeros, Proc. London Math. Soc. (2), 15 (1916), 314321.

25. S. Ramanujan, Some definite integrals, Messenger Math. 44 (1915), 1018; reprintedin [26].

26. S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, P. V. SeshuAiyar, and B. M. Wilson, Eds.), Cambridge Univ. Press, Cambridge, UK, 1927; reprintedby Chelsea, New York, 1962.

27. A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tiddskr. 26 (1944), 7178.

kadell k w j - a simple proof of an aomoto-type extension of askey's last conjectured selberg...

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