a continuous extension of a q-analogue of the 9-j symbols and its orthogonality

Advances in Applied Mathematics 46 (2011) 467–480

Contents lists available at ScienceDirect

Advances in Applied Mathematics

www.elsevier.com/locate/yaama

A continuous extension of a q-analogue of the 9- j symbolsand its orthogonality

Mizan Rahman 1

School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Available online 30 October 2010

Dedicated to our dear friend Dennis Stantonon the occasion of his 60th birthday

MSC:33D4533D50

Keywords:q-Racah and Askey–Wilson polynomials:q-analogue of 9- j symbolsBalanced and very-well-poisedhypergeometric seriesq-IntegralsOrthonormal functions in 2 continuousvariables

A q-analogue of Wigners’ 9- j symbols was found by the author ina recent paper where their orthogonality was also established. Inthis work we introduce a continuous version of these functionsand prove their orthogonality with respect to a 2-dimensionalextension of the Askey–Wilson weight.

© 2010 Published by Elsevier Inc.

1. Introduction

The orthonormality relation for q-Racah polynomials, see [3,8] or [11],

Wn(x) = Wn(x,a,b, c, N|q)

:= 4φ3

[q−n,abqn−1,q−x,acqx−1

a,abcqN−1,q−N ;q,q

](1.1)

E-mail address: [email protected] This work was supported by the NSERC Grant #A6197.

0196-8858/$ – see front matter © 2010 Published by Elsevier Inc.doi:10.1016/j.aam.2009.11.016

http://dx.doi.org/10.1016/j.aam.2009.11.016

http://www.ScienceDirect.com/

http://www.elsevier.com/locate/yaama

mailto:[email protected]

http://dx.doi.org/10.1016/j.aam.2009.11.016

468 M. Rahman / Advances in Applied Mathematics 46 (2011) 467–480

(note the slight difference in the notation for the Wn ’s compared to the standard one, see [8] or [3])is

N∑x=0

√hmhnρ(x)Wm(x)Wn(x) = δm,n, (1.2)

where the weight function is

ρ(x) = 1 − acq2x−1

1 − acq−1

(acq−1,a,abcqN−1,q−N )x

(q, c,q1−N/b,acqN)x(q/ab)x, (1.3)

and the normalization constant

hn = (b, c)N

(ab,ac)NaN 1 − abq2n−1

1 − abq−1

(abq−1,a,abcqN−1,q−N)n

(q,b,q1−N/c,abqN)n(q/ac)n. (1.4)

Askey and Wilson [4] found the following continuous extension of the discrete polynomials in (1.1):

rn(x) = rn(x;a,b, c,d|q)

:= 4φ3

[q−n,abcdqn−1,aeiθ ,ae−iθ

ab,ac,ad;q,q

], (1.5)

where x = cos θ , 0 � θ � π , and max(|a|, |b|, |c|, |d|, |q|) < 1. The orthonormality relation for thesepolynomials is

1∫−1

√Nm Nn w(x)rm(x)rn(x)dx = δm,n, (1.6)

where the continuous weight function w(x) is defined by

w(x) = w(x;a,b, c,d|q)

= h(x;1,−1,q1/2,−q1/2)(1 − x2)−1/2

h(x;a,b, c,d)(1.7)

with

h(x;a1,a2, . . . ,ak) =k∏

i=1

h(x;ai),

h(x;a) = (aeiθ ,ae−iθ )

∞, (1.8)

and the normalization constant is

Nn = κ−1(a,b, c,d)(1 − abcdq2n−1)

−1

(abcdq−1,ab,ac,ad)na−2n, (1.9)

(1 − abcdq ) (q, cd,bd,bc)n

M. Rahman / Advances in Applied Mathematics 46 (2011) 467–480 469

with

κ(a,b, c,d) = 2π(abcd)∞(q,ab,ac,ad,bc,bd, cd)∞

, (1.10)

see also [8,11] and [14].The 4φ3 series in (1.1) and (1.5) are special cases of the basic hypergeometric series

r+1φr(a1,a2, . . . ,ar+1;b1,b2, . . . ,br;q, z) ≡ r+1φr

[a1,a2, . . . ,ar+1

b1, . . . ,br;q, z

]

=∞∑

n=0

(a1,a2, . . . ,ar+1;q)n

(q,b1, . . . ,br;q)nzn, (1.11)

where the q-shifted factorials are defined by

(a;q)n ={

1, if n = 0,

(1−a)(1 − aq) . . . (1−aqn−1), if n = 1,2, . . . ,

(a1, . . . ,ak;q)n =k∏

�=1

(a�;q)n, (1.12)

and |z| < 1 if the series in (1.11) does not terminate. This series is balanced if qa1a2 . . .ar+1 = b1 . . .br ,and well poised if qa1 = a2b2 = · · · = ar+1br . If a2b2 = −a3b3 = qa1/2

1 in a well-poised series, as wellas the parameter z satisfies the so-called balancing condition

(a4a5 . . .ar+1)z = (±(a1q)1/2)r−3, (1.13)

then the r+1φr series above is called a very-well-poised series, and is usually denoted by

r+1Wr(a1;a4,a5, . . . ,ar+1;q, z). (1.14)

Note that the 4φ3 series in (1.1) and (1.5) are both balanced and terminating. The series that we shalluse in this paper are either balanced or very well poised. For the sake of brevity we shall drop the“; q” part from the q-shifted factorials throughout this paper. We shall also assume that |q| < 1, sincethis condition will be necessary in ascertaining the convergence of the infinite products like

(a;q)∞ =∞∏

n=0

(1 − aqn), (1.15)

which will be used in this work.In recent years there have been many multidimensional extensions of single-variable orthogonal

polynomials, including some that generalize the q-Racah polynomials (1.1). See [6,9,15,16] and [17].The precursor of the present work is [10] where M. Hoare and the author were able to derive a2-variable extension of the Krawtchouk polynomials from the 9- j symbols of quantum angular mo-mentum theory, see [5] or [7], whose orthogonality is assured by their physical properties. This ledus to the search for a q-analogue of the 9- j symbols that has the orthogonality property. Unbeknownto the author such analogues already existed in the literature as early as 1971 [2] (thanks to HjalmarRosengren for pointing this out to me), and in a more expanded form recently in [1]. However, itis the opinion of the present author that these analogues do not have the orthogonality property —


a crucial term appears to be missing in the q-analogue of the 9- j symbols in [2]. The author intro-duced a different analogue in [13], and proved its orthogonality, see also [12].

The main objective of this work is to give a continuous version of these polynomials in the samesense as (1.5) is a continuous version of (1.1), that may be considered as a 2-dimensional extensionof the Askey–Wilson polynomials. To that end it is important (though not essential) to transform theq-analogue of the 9- j symbol given in [12] and [13] to a more convenient form:

Rτm,n(x, y|q) = Cτ

m,n(x, y|q)∑

�

1 − q1−2N+2�/bcd

1 − q1−2N/bcd

(q1−2N/bcd,a)�

(q,q2−2N/abcd)�

× (qn−N ,q1−N−n/bd,qy−N ,q1−N−y/cd,q2−N/bcd)�

(q2−N−n/bcd,q1−N+n/c,q2−N−y/bcd,q1−N+y/b,q−N)�

{q

aτ

(d

bc

)1/2}

× W�

(x;a,q2−2N/abcd,b, N − y|q)

× W�

(m;a,q2−2N/abcd, c, N − n|q)

× Wn(y;d,b, c, N − �|q), (1.16)

where

Cτm,n(x, y|q) = (q)N (abcdq−1)2N(b)N−y(bcdqN−1)y(c)N−n(bcdqN−1)n

(bcdqN−1)Nτ y−N

×{

1 − abq2x−1

1 − abq−1

(abq−1,a,q1−N−y/cd,qy−N)x

(q,b,abcdqN+y−1,abqN−y)x

(bcdq2N−1)x

× 1 − acq2m−1

1 − acq−1

(acq−1,a,q1−N−n/bd,qn−N)m

(q, c,abcdqN+n−1,acqN−n)m

(bcdq2N−1)m

× 1 − cdq2y−1

1 − cdq−1

(cdq−1,d)y

(q, c)ydN−y

× 1 − bdq2n−1

1 − bdq−1

(bdq−1,d)n

(q,b)nd−n

× ((abcdq−1, cd

)N+y(q,ab)N−y

(abcdq−1,bd

)N+n(q,ac)N−n

)−1}1/2

, (1.17)

with τ = ±1. In this form the duality of the pairs (x,m) as well as (y,n) is expressed as explicitlyas possible. In Section 2 we will give a sketch of the steps that lead to this form. In Section 3 weshow how a close examination of the relation between (1.1) and (1.5) can guide us to the followingcontinuous version of the series in (1.16):

W τm,n(x, y|q) = Nm,n(x, y|q)

∞∑�=0

1 − αβ2γ q2�−1

1 − αβ2γ q−1

(αβ2γ q−1,αγ )�

(q, β2)�

× (αβeiφ,αβe−iφ,βγ /ε)�

(βγ e−iφ,βγ eiφ,αβε)�

(ε

ατ√

αγ

)�

× r�

(x;α,βeiφ,βe−iφ,γ |q)

rn(

y;ε, δ,αβq�,q1−�/βγ |q)

× 4φ3

[q−�,αβ2γ q�−1,q−m,αγ δε qm

n βγ q−n ;q,q

], (1.18)

αγ ,αβδq , ε


where x = cos θ , y = cosφ, 0 � θ , φ � π , and

Nm,n(x, y|q) = (q,qα2, β2, δε)∞2π(αβ2γ )∞

h(cosφ;βγ )

×{

1 − βγ δq2m−1

1 − βγ δq−1

(βγ δq−1,αβδqn, γ β2q−n−1/α,αγ )m

(q, γ q−n/α,αδqn+1/β,βδ/α)m

(q/αβ2γ

)m

× 1 − δα2q2n+1/βγ

1 − δα2q/βγ

(qδα2/βγ ,αq2/β2γ )n

(q,αβδ)n

(γ β2/αq2)n

× (αγ ,βδ/α,αq2/γ β2,αβδ)∞(βγ δ, δα2q2/βγ )∞

× csc θ csc φ(e2iθ , e−2iθ , e2iφ, e−2iφ)∞h(cos θ;α,βeiφ,βe−iφ,γ )h(cosφ;β/α,ε, ε, δ)

}1/2

. (1.19)

Note that the series over � in (1.18) is an infinite series and hence it must satisfy the conver-gence condition |ε/ατ

√αγ | < 1, which may be modified, if necessary, by analytic continuation. It

may be worth emphasizing that although we claim W τm,n(x, y|q) to be a 2-variable extension of the

Askey–Wilson polynomial, it itself is not a polynomial; first, because these are orthonormal func-tions; second, because the weight function is not factorizable in the general case, as is the case inone-dimensional AW polynomial. The main result of this paper is the orthogonality relation

1∫−1

1∫−1

W τm,n(x, y|q)W τ

m′,n′(x, y|q)dx dy = δm,m′δn,n′ . (1.20)

For the convergence of this integral it is sufficient to assume that

max(|α|, |β|, |γ |, |δ|, |ε|) < 1. (1.21)

Unfortunately, however, for the orthogonality relation (1.20) to hold we need to impose the morerestrictive condition

ε = qα/β and q < |β/α| < 1. (1.22)

It is not unlikely that there exists a different extension of the 9- j symbols that is more generalthan this one, but so far we were unable to find it. As we shall see in Sections 3 to 6 even with therestrictions (1.22) the proof of (1.20) is not entirely straightforward.

2. Derivation of (1.16)

The q-analogue of the 9- j symbol given in (1.25) of [12], which is equivalent to the one givenearlier in [13], is

Rτm,n(x, y|q) = Am,n(x, y|q)

N−y∑�=0

1 − bcdq2�+2y−1

1 − bcdq2y−1

(bcdq2y−1,abcdqN+y−1)�

(q,q1−N+y/a)�

× (b, cqy−n,qy−N)�2y y+n N+y

{τqn+1(d/bc)1/2/a

}�

(cdq ,bdq ,bcdq )�


× W�

(x;b, cdq2y,a, N − y|q)

W y−n+�

(m; c,bdq2n,a, N − n|q)

× W�

(n;b, cdq2y,q2−2n/bd,n − y − 1|q)

, (2.1)

where

Am,n(x, y|q) = (q)N(a)N−y(bcd)2y(q−y)nq(n2)(−qy−N)n

(abcdq−1)N+n(bcd)N+y(cd)2y(bd)y+n(q−N)n

×{

1 − abq2x−1

1 − abq−1

(abq−1,b,abcdqN+y−1,qy−N)x

(q,a,q1−N−y/cd,abqN−y)x

(q1−2y/bcd

)x

× (abcdq−1, cd)N+y

(q,ab)N−y

1 − cdq2y−1

1 − cdq−1

(cdq−1, c,d)y

(q)y(bc)N−y

× 1 − acq2m−1

1 − acq−1

(acq−1, c,abcdqN+n−1,qn−N)m

(q,a,q1−N−n/bd,acqN−n)m

(q1−2n/bcd

)m

× (abcdq−1,bd)N+n

(q,ac)N−n

1 − bdq2n−1

1 − bdq−1

(bdq−1,b)n

(q,d)ndn

}1/2

. (2.2)

First, by Sears’ transformation formula [8, III.15] for balanced terminating 4φ3 series, we have

W�

(x;b, cdq2y,a, N − y|q)

= 4φ3

[q−x,abqx−1,q−�,bcdq�+2y−1

b,qy−N ,abcdqN+y−1 ;q,q

]

= (a,q1−N−y/cd)x

(b,abcdqN+y−1)x

(bcdqN+y−1)x

W N−y−�

(x;a,q2−2N/abcd,b, N − y|q)

, (2.3)

W y−n+�

(m; c,bdq2n,a, N − n|q)

= 4φ3

[q−m,acqm−1,qn−y−�,bcdqn+y+�−1

qn−N , c,abcdqN+n−1 ;q,q

]

= (a,q1−N−n/bd)m

(c,abcdqN+n−1)m

(bcdqN+n−1)m

W N−y−�

(m;a,q2−2N/abcd, c, N − n|q)

. (2.4)

Now, by applying Watson’s transformation formula [8, III.18] twice, we find that

W�

(n;b, cdq2y,q2−2n/bd,n − y − 1|q)

= 4φ3

[q−n,q1−n/d,q−�,bcdq�+2y−1

b, cqy−n,qy−n+1 ;q,q

]

= (bcdqy+�−1,q−y−�)n

(bcdqy−1,q−y)n8φ7

[bcdqy−2,q

√·,−q√·,q−n,bdqn−1,√·,−√·,bcdqy+n−1, cqy−n,

cdqy−1,bcdq�+2y−1,q−�

b,q−y−�,bcdqy+�−1 ;q,q1−y/d

]


= (bcdqy+�−1,q−y−�,d)n

(cqy−n,b,q−y)n(−1/d)nq−(n

2) 4φ3

[q−n, cdqn−1,q−y, cdqy−1

d,bcdqy+�−1,q−y−� ;q,q

]

= (d,bdqy+�−1,q−y−�)n

(b,q−y, cqy−n)n(−1/d)nq−(n

2)Wn(y;d,b, c, y + �|q). (2.5)

The next step is to substitute (2.3), (2.4) and (2.5) into (2.1), reverse the order of �-summation (i.e.replace � by N − y − �), then simplify. The result is (1.16).

3. A continuous q-analogue

As far as the series part of (1.16) is concerned one can easily construct a formal analogue byextending the analogy between the q-Racah polynomial (1.1) in a discrete variable x and the Askey–Wilson polynomial (1.5) in the continuous variable x = cos θ . Accordingly, we replace q−x and abqx−1

by αe−iφ and αeiφ , respectively, as well as q−y and cdqy−1 by δe−iφ and δeiφ , in (1.16). Recognizingthat the individual parameters a, b, c, d and N need to be adjusted accordingly, let us set

a = αγ , b = qα/γ , c = qδ/ε, d = δε, q−N = αβδ, (3.1)

so that

qy−N = αβeiφ, q1−N−y/cd = αβe−iφ. (3.2)

Then the series part of (1.14) changes to

∞∑�=0

1 − αβ2γ q2�−1

1 − αβ2γ q−1

(αβ2γ q−1,αγ ,αβδqn, βγ q−n/ε,αβeiφ,αβe−iφ,βγ /δ)�

(q, β2, βγ q−n/δ,αβεqn, βγ e−iφ,βγ eiφ,αβδ)�

×(

ε

τα√

αγ

)�

r�

(x;α,γ ,βeiφ,βe−iφ |q)

rn(

y; δ, ε,αbq�,q1−�/βγ |q)

× 4φ3

[q−�,αβ2γ q�−1,q−m,

αγ δqm

εαγ ,αβδqn, βγ q−n/ε

;q,q

]. (3.3)

Since the �-series has the structure of a very-well-poised, nonterminating and unbalanced 10φ9 seriesthat cannot be transformed to another 10φ9 series or anything else in a simple way, it is unlikely that(3.3) would be a suitable analogue. What we need to do is reduce the size. This can be done in twoways. Apply [8, III.15] on the 4φ3 series above to knock out the n-dependent terms in the very-well-poised part of the sum, or apply the same transformation to rn(y) part to do essentially the samething. The first would force one to replace αγ qm or εq−m/δ by something like q−r , and assume rto be a nonnegative integer. Since the second alternative does not require any such replacement, yetgives us the same result in the end we chose to take

rn(

y; δ, ε,αβq�,q1−�/βγ |q) = (qε/βγ ,αβε)n

(αβδ,qδ/βγ )n(δ/ε)n (βγ /ε,αβδ)�

(αβε,βγ /δ)�

(βγ q−n/δ,αβεqn)�

(αβδqn, βγ q−n/ε)�

× rn(

y;ε, δ,αβq�,q1−�/βγ |q). (3.4)

Ignoring the part dependent only on n leads us to the series part of (1.18).


4. Orthogonality in x

In this section we shall do the integration over x in (1.20) by using (1.6), since x appears only inthe r�(x;α,βeiφ,βe−iφ,γ |q) part. Note that

Nm,n(x, y|q)Nm′,n′(x, y|q)

= (q,qα2, β2, δε)2∞h2(cosφ;βγ )

4π2(αβ2γ )2∞

√hm,nhm′,n′

(αγ ,βδ/α,αβδ,αq2/γ β2)∞(βγ δ, δα2q2/βγ )∞

× w(x;α,βeiφ,βe−iφ,γ |q)

w(y;β/α,ε, ε, δ|q) (4.1)

where

hm,n = 1 − βγ δq2m−1

1 − βγ δq−1



(q/αβ2γ

)m

× 1 − δα2q2n+1/βγ

1 − δα2q/βγ

(qδα2/βγ ,αq2/β2γ )n

(q,αβδ)n

(γ β2/αq2)n

, (4.2)

and assuming (1.20) to hold.By (1.6)

1∫−1

w(x;α,βeiφ,βe−iφ,γ |q)

r�

(x;α,βeiφ,βe−iφ,γ |q)

r�′(x;α,βeiφ,βe−iφ,γ |q)

dx

= 2π(αβ2γ )∞(q,αγ ,β2)∞h(cosφ;αβ,βγ )

× 1 − αβ2γ q−1

1 − αβ2γ q2�−1

(q, βγ e−iφ,βγ eiφ,β2)�

(αβ2γ q−1,αβeiφ,αβe−iφ,αγ )�α2�δ�,�′ . (4.3)

So

1∫−1

W τm,n(x, y|q)W τ

m′,n′(x, y|q)dx

= (qα2, δε)2∞(q, β2, βδ/α,αβδ,qε/βγ )∞h(cosφ;βγ )

2π(αβ2γ ,βγ δ,βδε2/γ )∞h(cosφ;αβ)

× (hm,nhm′,n′)1/2 w(y;β/α,ε, ε, δ|q)

×∑

�

1 − αβ2γ q2�−1

1 − αβ2γ q−1

(αβ2γ q−1,αγ ,αβeiφ,αβe−iφ,βγ /ε,βγ /ε)�

(q, β2, βγ e−iφ,βγ eiφ,αβε,αβε)�

(ε2/αγ

)�

× rn(

y;ε, δ,αβq�,q1−�/βγ |q)rn′

(y;ε, δ,αβq�,q1−�/βγ |q)

× 4φ3

[q−�,αβ2γ q�−1,q−m,

αγ δqm


;q,q

]4φ3

[q−�,αβ2γ q�−1,q−m′

,αγ δqm′

ε

αγ ,αβδqn′, βγ q−n′

/ε;q,q

], (4.4)

with ε = qα/β .


5. Integration over y

By [12, (A.6)],

4φ3

[q−�,αβ2γ q�−1,q−m,

αγ δqm


;q,q

]4φ3

[q−�,αβ2γ q�−1,q−m′

,αγ δqm′

ε

αγ ,αβδqn′, βγ q−n′

/ε;q,q

]

=∑

k

(q−�,αβ2γ q�−1,q−m,αγ δqm/ε)k

(q,αγ ,αβδqn, βγ q−n/ε)kqk

×∑

j

(q−k,αβ2γ qk−1,q−m′,αγ δqm′

/ε) j

(q,αγ ,αβδqn′, βγ q−n′

/ε) jq j

× 4φ3

[q− j,q1−k/αγ ,q1−n−k/αβδ, εq1+n−k/βγ

q2−k− j/αβ2γ ,qm+1−k, εq1−m−k/αδγ;q,q

]. (5.1)

Hence the series on the right-hand side of (4.4) becomes

∑k

(q−m,αγ δqm/ε)k

(q,αγ ,αβδqn, βγ q−n/ε)k

∑j


/ε) j


/ε) jq j

× 4φ3

[q− j,q1−k/αγ ,q1−n−k/αβδ, εqn+1−k/βγ

q2−k− j/αβ2γ ,qm+1−k, εq1−m−k/αγ δ;q,q

]

×n∑

r=0

(q−n,αδεqn/γ , εeiφ, εe−iφ)r

(q, δε,αβε,qε/βγ )rqr

n′∑s=0

(q−n′,αδεqn′

/γ , εeiφ, εe−iφ)s

(q, δε,αβε,qε/βγ )sqs

× (αβ2γ )2k(αγ ,βγ q−r/ε,βγ q−s/ε,αβeiφ,αβe−iφ)k

(β2,αβεqr,αβεqs, βγ eiφ,βγ e−iφ)k

(−ε2qr+s

αγ

)k

q−(n2)

× 8W7(αβ2γ q2k−1;αγ qk,αβqkeiφ,αβqke−iφ,βγ qk−r/ε,βγ qk−s/ε;q, ε2qr+s−k/αγ

),

(5.2)

where we use the 8W7-notation for a very-well-poised 8φ7 series, see [8]. For convergence of thisseries we need to assume

∣∣ε2q−m/αγ∣∣ < 1, (5.3)

which, for small enough m, can be made to satisfy. However, there are transformation formulas fornonterminating 8W7 series that we can use to resolve this convergence difficulty in two differentways. We can use [8, III.24] to get

8W7(αβ2γ q2k−1;αγ qk,αβqkeiφ,αβqke−iφ,βγ qk−r/ε,βγ qk−s/ε;q, ε2qr+s−k/αγ

)

= (αβ2γ q2k,αγ qk, βε2eiφqr+s/γ ,βε2e−iφqr+s/γ ,βεqr+1/α,βεqs+1/α)∞(βγ qkeiφ,βγ qke−iφ,αβεqk+r,αβεqk+s, β2ε2qr+s+1/αγ ,ε2qr+s−k/αγ )∞× 8W7

(β2ε2qr+s/αγ ;ε2qr+s+1−k/αγ ,βeiφ/α,βe−iφ/α,βεqr/γ ,βεqs/γ ;q,αγ qk),

(5.4)

or use the q-integral representation [8, (2.10.19)] to get


8W7(αβ2γ q2k−1;αγ qk, . . . , βγ qk−s/ε;q, ε2qr+s−k/αγ

)

= q−k

α2(1 − q)

(γ /α,αγ qk,αβ2γ q2k,αε2qr+s/γ )∞h(y;αβqk, β/a)

(q,q−k/α2,α2qk+1, β2qk,αβεqk+r,αβεqk+s)∞h(y;βγ qk)

×α2qk∫1

(qt, tq1−k/α2, βεtqr/α,βεtqs/α)∞(γ t/α, tε2qr+s−k/αγ )∞h(y;βt/α)

dqt. (5.5)

We choose to use (5.5) since, in order to do the integration over y, we would need to convert the8φ7 series on the right-hand side of (5.4) to this form anyway.

Substituting (5.5) into (5.2) we find that the y-integral boils down to evaluating

1∫−1

w(

y;βt/α,εqr, εqs, δ|q)dy

which is, by [8, (6.1.11)],

2π(tβδε2qr+s/α)∞(q, βεtqr/α,βεtqs/α,βδt/α,ε2qr+s, δεqr, δεqs)∞

, (5.6)

so that the q-integral in (5.5) reduces to

α2qk∫1

(qt, tq1−k/α2, tβδε2qr+s/α)∞(γ t/α, tε2qr+s−k/αγ ,βδt/α)∞

dqt

= α2(1 − q)qk (q,q−k/α2,α2qk+1, βδε2qr+s/γ ,βγ δqk, ε2qr+s)∞(γ /α,βδ/α,ε2qr+s−k/αγ ,αγ qk,αβδqk,αε2qr+s/γ )∞

, (5.7)

by [8, (2.10.18)]. Hence, from (4.4), (5.2), (5.5)–(5.7), we find that

Im′,n′m,n :=

1∫−1

1∫−1

W τm,n(x, y|q)W τ

m′,n′(x, y|q)dx dy

= (hm,nhm′,n′)1/2∑

k

(q−m,αγ δqm/ε,αβδ)kqk

(q,αβδqn, βγ δ,βγ q−n/ε)k

×∑

j


/ε) j


/ε) jq j (5.8)

× 4φ3

[q− j,q1−k/αγ ,q1−n−k/αβδ, εqn+1−k/βγ

q2−k− j/αβ2γ ,qm+1−k, εq1−m−k/αγ δ;q,q

]

×∑

r

∑s

(q−n,αδεqn/γ )r(q−n′,αδεqn′

/γ )s(ε2/αγ )r+s

(q,qε/βγ )r(q,qε/βγ )s(βδε2/γ )r+sqr+s

× (βγ q−r/ε,βγ q−s/ε)k

(αγ q1−r−s/ε2)k, (5.9)

where we have used ε = qα/β to simplify the coefficient, but not in the series.


6. Orthogonality

We now proceed to prove the orthogonality relation (1.18). First note that the double series over rand s in (5.8) can be written as

(βγ /ε)k

∑s

(q−n′,αδεqn′

/γ )s

(q, βδε2/γ )sqs

× 3φ2

[q−n,αδεqn/γ , ε2qs−k/αγ

βδε2qs/γ , εq1−k/βγ;q,q

]. (6.1)

The 3φ2 series above is balanced and terminating, so by [8, II.12] it has the sum

(αβδqk, βεqs−n/α)n

(βγ qk−n/ε,βδε2qs/γ )n. (6.2)

It is precisely at this point where we need to impose the condition (1.22), otherwise there is noorthogonality. We will assume that n′ � n, so that

(βεqs−n/α

)n = (

qs+1−n)n =

{0 if s < n,

�= 0 if s � n.(6.3)

Replacing the summation index s by s + n the series (6.1) reduces to

(βγ /ε)k(q−n′

,αδεqn′/γ ,αβδqk)n

(βγ qk−n/ε)n(βδε2/γ )2nqn

× 2φ1

[qn−n′

,αδεqn+n′/γ

αβεq2n+1/γ;q,q

]. (6.4)

By [8, II.6] the sum of the 2φ1 series above is

(q1+n−n′)n′−n

(αβδεq2n+1/γ )n′−n,

which vanishes unless n′ = n. So the expression in (6.4) becomes

= (βγ /ε)k

(q−n, αδεγ qn,αβγ qk)n

(βγ qk−n/ε)n(qαδε/γ )2nqnδn,n′ . (6.5)

Substitution of (6.5) into (5.8) followed by some simplification and use of (4.2) gives us

Im′,n′m,n = δn,n′(hmhm′)1/2

×∑

k

(q−m, βγ δqm−1)k

(q, βγ δ)kqk

×∑

j

(q−k,αβ2γ qk−1,q−m′, βγ δqm′−1) j

(q,αγ ,αβδqn, β2γ q−n−1/α) jq j

× 4φ3

[q− j,q1−k/αγ ,q1−n−k/αβδ,αqn+2−k/β2γ

q2−k− j/αβ2γ ,qm+1−k,q2−m−k/βγ δ;q,q

], (6.6)


where

hm = 1 − βγ δq2m−1

1 − βγ δq−1



(q/αβ2γ

)m. (6.7)

However, by [8, III.15],

4φ3

[q− j,αqn+2−k/β2γ ,q1−k/αγ ,q1−n−k/αβδ

qm+1−k,q2−m−k/βγ δ,q2−k− j/αβ2γ;q,q

]

= (α2qn+1, βq−m−n/αδ) j

(αβ2γ qk−1,q2−m−k/βγ δ) j4φ3

[q− j,αqn+2−k/β2γ ,αγ qm+1,αβδqm+n

qm+1−k,α2qn+1,αδqm+n+1− j/β;q,q

], (6.8)

which, on substitution in (6.6), leads to the following:

Im′,n′m,n = δn,n′(hmhm′)1/2

∑k

∑j

∑�

(q−m)k−�(βγ δqm−1)k− j

(γ β2q−n−1/α)k−�(q)k− j

× (βq−m−n/αδ) j−�

(q) j−�

(γ β2q−n−1/α)k(αγ qm,αβδqm+n)�

(βγ δ)k(q,α2qn+1)�

× (α2qn+1,q−m′, βγ δqm′−1) j

(αγ ,αβδqn, γ β2q−n−1/α) j

(βγ δqm−1) j−�

qk−m�

= δn,n′(hmhm′)1/2∑

k

∑j

∑�

(q−m)k+ j(βγ δqm−1)k(βq−m−n/αδ) j

(γ β2q−n−1/α)k+ j(q)k(q) j

× (γ β2q−n−1/α)k+ j+�(αγ qm,αβδqm+n)�

(βγ δ)k+ j+�(q,α2qn+1)�

× (α2qn+1,q−m′, βγ δqm′−1) j+�

(αγ ,αβδqn, γ β2q−n−1/α) j+�

(βγ δqm−1) j

qk+ j+�−m�. (6.9)

The sum over k is

3φ2

[q j−m, βγ δqm−1, γ β2q j+�−n−1/α

βγ δq j+�, γ β2q j−n−1/α;q,q

]

= (q1+ j−m+�,αδqn+1/β)m− j

(βγ δq j+�,αq2+n−m/β2γ )m− j, by [8, II.12], (6.10)

which vanishes unless m′ � � � m − j. Using of (6.10) in (6.9) gives us, on some simplification,

Im′,n′m,n = δn,n′(hmhm′)1/2 (q−m′

, βγ δqm′−1,αγ qm,αβδqm+n,αδqn+1/β)m

(αγ ,αβδqn,αq2+n−m/β2γ )m(βγ δ)2m

× qm−m2 ∑�

(qm−m′, βγ δqm+m′−1,αγ q2m,αβδq2m+n)�

(q, βγ δq2m,αγ qm,αβδqm+n)�q�(1−m)

× 3φ2

[q−m,q1−2m−�/βγ δ,q−m−n−�/α2

q1−2m−n−�/αβδ,q1−2m−�/αγ;q,q

]. (6.11)


This last 3φ2 series is also balanced, and has the sum

(q−nγ /α,αq1−m/βδ)m

(q1−2m−n−�/αβδ,αγ qm+�)m

= (γ q−n/α,βδ/α)m

(αβδqm+n,αγ qm+�)m

(α2qm+n)m

qm� (αβδqm+n)�

(αβδq2m+n)�. (6.12)

Hence

Im′,n′m,n = δn,n′(hmhm′)1/2 (q−m′

, βγ δqm′−1,αδqn+1/β,γ q−n/α,βδ/α)m

(αγ ,αq2+n−m/β2γ ,αβδqn)m(βγ δ)2m

× (α2qn+1)m

2φ1

[qm−m′

, βγ δqm+m′−1

βγ δq2m ;q,q

]. (6.13)

Since

2φ1[ ] = (q1+m−m′)m′−m

(βγ δq2m)m′−m

(βγ δqm+m′−1)m′−m

= δm,m′ , (6.14)

by [8, II.6], and

(q−m, βγ δqm−1,αδqn+1/β,γ q−n/a, βδ/α)m

(αq2+n−m/β2γ ,αβδqn,αγ )m(βγ δ)2m

(α2qn+1)m

= 1 − βγ δq−1

1 − βγ δq2m−1

(q, βδ/α,αδqn+1/β,γ q−n/α)m

(βγ δq−1,αγ ,β2γ q−n−1/α,αβδqn)m

(αβ2γ q−1)m

= h−1m , by (6.7), (6.15)

it follows that

Im′,n′m,n = δm,m′δn,n′ ,

which completes the proof of (1.20), subject to the conditions (1.21) and (1.22).

Acknowledgments

We would like to thank the two referees for having done an excellent job giving a thorough readingof the paper that enabled them to suggest some interesting new ideas.

References

[1] S. Alisauskas, The multiple sum formulas for 9- j and 12- j coefficients of SU(2) and uq(2), arXiv:math/9912142v5[math.QA], 2008.

[2] S. Alisauskas, A.P. Jucys, Weight lowering operators and the multiplicity-free isoscalar factors for the group R5, J. Math.Phys. 12 (1971) 594–605.

[3] R. Askey, J.A. Wilson, A set of polynomials that generalize Racah coefficients or 6- j symbols, SIAM J. Math. Soc. 10 (1979)1008–1016.

[4] R. Askey, J.A. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc.(1985), #319.

[5] L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Mechanics, Theory and Applications, Encyclopedia Math. Appl.,vol. 8, Addison–Wesley, Reading, 1981.


[6] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge, UK, 2001.[7] A.R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd edn., Princeton University Press, Princeton, NJ, 1960.[8] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd edn., Cambridge University Press, Cambridge, UK, 2004.[9] G. Gasper, M. Rahman, Some systems of multivariable orthogonal q-Racah polynomials, Ramanujan J. 13 (2007) 389–405.

[10] M. Hoare, M. Rahman, A probabilistic origin for a new class of bivariate polynomials, SIGMA Symmetry Integrability Geom.Methods Appl. 4 (2008) 089, arXiv:0812.3879v1 [math.CA].

[11] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, UK,2005.

[12] M. Rahman, An explicit polynomial expression for a q-analogue of the 9- j symbols, Canad. J. Math. (2009), in press.[13] M. Rahman, A q-analogue of the 9- j symbols and their orthogonality, J. Approx. Theory 161 (2009) 239–258.[14] M. Rahman, S.K. Suslov, The Pearson equation and the q-beta integrals, SIAM J. Math. Anal. 15 (1994) 646–693.[15] M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey-tableau—discrete families, J. Math. Phys. 32 (1991)

2337–2342.[16] J.F. van Diejen, J.V. Stokman, Multivariable q-Racah polynomials, Duke Math. J. 91 (1998) 89–136.[17] Y. Xu, On discrete orthogonal polynomials of several variables, Adv. in Appl. Math. 33 (2004) 615–632.

a continuous extension of a q-analogue of the 9-j symbols and its orthogonality

Documents