Journal of Combinatorial Theory, Series A

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1 Journal of Combinatorial Theory, Series A 118 (2011) Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series A wwwelseviercom/locate/jcta Bailey s well-poised 6 ψ 6 -series implies the Askey Wilson integral Wenchang Chu a,1, Xinrong Ma b,2 a Hangzhou Normal University, Institute of Combinatorial Mathematics, Hangzhou , PR China b Suzhou University, Department of Mathematics, Suzhou , PR China article info abstract Article history: Received 10 July 2009 Available online 5 November 2009 Keywords: Askey Wilson integral Bailey s identity of well-poised 6 ψ 6 -series Bailey s fundamental identity of bilateral well-poised 6 ψ 6 -series is utilized to present shorter proofs for the two important q- beta integrals discovered by Askey and Wilson (1985) [4] and Askey (1987) [3] Another rather general q-beta integral containing many extra parameters is similarly derived from Chu s extended Karlsson Minton-type identity for a bilateral well-poised 6+2n ψ 2n+6 - series 2009 Elsevier Inc All rights reserved Bailey s identity of bilateral well-poised 6 ψ 6 -series is one of the deepest results in the theory of basic hypergeometric series, which can be reproduced as follows Letting a, b, c, d and λ be five complex parameters subject to qλ 2 /abcd < 1, there holds the summation formula [5] (see also [8], [10, II-33] and [19, 71]) [ q λ, q λ, a, b, c, d 6ψ 6 q; qλ 2 ] λ, λ, qλ/a, qλ/b, qλ/c, qλ/d abcd (1a) k 1 q 2k λ 1 λ (a, b, c, d; q) k (qλ/a, qλ/b, qλ/c, qλ/d; q) k ( qλ 2 ) k (1b) abcd (q, qλ,q/λ, qλ/ab, qλ/ac, qλ/ad, qλ/bc, qλ/bd, qλ/cd; q) (qλ/a, qλ/b, qλ/c, qλ/d, q/a, q/b, q/c, q/d, qλ 2 /abcd; q) (1c) addresses: chuwenchang@unileit (W Chu), xrma@sudaeducn (X Ma) 1 Current address: Dipartimento di Matematica, Università del Salento, Lecce-Arnesano, PO Box 193, Lecce 73100, Italy 2 Partially supported by NSF of China (Grant No ) /$ see front matter 2009 Elsevier Inc All rights reserved doi:101016/jjcta

2 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) where the q-shifted factorial with 0 < q < 1isdefinedby ( (x; q) ) 1 xq k and (x; q) n (x; q) (q n for n Z x; q) k0 whose multi-parameter form is compactly abbreviated to (α,β,,γ ; q) n (α; q) n (β; q) n (γ ; q) n The purpose of this note is to show that Bailey s identity of 6 ψ 6 -series implies, surprisingly, both the important q-beta integral discovered by Askey and Wilson [4] and its reversal due to Askey [3, Eq 311] Furthermore, we shall derive another general q-beta integral formula by means of the extended identity [7, Theorem 2] of Karlsson Minton type for bilateral well-poised 6+2n ψ 2n+6 -series We remark that the same approach presented here has been employed in [1, 108] to evaluate q-beta integrals using Ramanujan s formula of bilateral 1 ψ 1 -series 1 The Askey Wilson integral Define the h-function by the modified Jacobi theta function h(x; λ) ( λe iθ,λe iθ ; ) q ( 1 2q k λx + q 2k λ 2) where x cos θ and its multi-parameter form k0 h(x; α,β,,γ ) h(x; α)h(x; β) h(x; γ ) Then the q-beta integral due to Askey and Wilson [4, Theorem 21] (see [1, Chapter 10] and [10, Chapter 6]) reads as follows Theorem 1 (Askey and Wilson [4, Theorem 21]) For abcd/q < 1, there holds π 0 h(cos 2θ; 1) h(cos θ; a, b, c, d) dθ 2π(abcd; q) (q, ab, ac, ad, bc, bd, cd; q) This integral plays a central role in the theory of orthogonal polynomials Different proofs can be found in [3,6,11 13,15] We refer to [14,16 18,20] for extensions and to [2] as well as the notes at the end of [10, Chapter 6] for multiple integrals related to constant term identities Proof We shall show that Theorem 1 follows directly from Bailey s identity of bilateral 6 ψ 6 -series, which is shorter and more transparent than the original proof due to Askey and Wilson [4] In fact, performing the replacements in Eq (1) λ qz 2, a qz/a, b qz/b, c qz/c, d qz/d; and then multiplying across that equation by (1 z 2 )(1 qz 2 ) (1 az)(1 bz)(1 cz)(1 dz)

3 242 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) we may write the right member of the resulting equation as f (z) (z 2, 1/z 2 ; q) (q, ab, ac, ad, bc, bd, cd; q) (az, a/z, bz, b/z, cz, c/z, dz, d/z; q) (abcd/q; q) (2) While the corresponding left member reads as (1 z 2 )(1 qz 2 [ 3 3 ) ] q 2 z, q 2 z, qz/a, qz/b, qz/c, qz/d q; abcd (1 az)(1 bz)(1 cz)(1 dz) 6 ψ 6 q z, q 2 z, qaz, qbz, qcz, qdz q provided that abcd/q < 1 for convergence Denoting by k the summation index for the last 6 ψ 6 - series and then replacing it by k k 1 for the terms with negative integer indices, we get an alternative expression f (z) ( )( 1 z 2 1 q 1+2k z 2) (qz/a, qz/b, qz/c, qz/d; q) ( ) k k abcd (az, bz, cz, dz; q) k+1 q + ( )( 1 z 2 1 q 1+2k z 2) (q/az, q/bz, q/cz, q/dz; q) ( ) k k abcd (3) (a/z, b/z, c/z, d/z; q) k+1 q Therefore f (z) is regular within 0 < z < and can be expanded into a Laurent series at z 0with the constant term equal to [ ] ( ) k z 0 abcd 2 f (z) 2 q 1 abcd/q Let z e iθ with θ R and then integrate f (z) over π θ π According to Eq (3), it is trivial to see that π ( f e iθ ) dθ 2π [ z 0] 4π f (z) 1 abcd/q π Instead, we deduce from (2) alternatively that π π f ( e iθ ) dθ (q, ab, ac, ad, bc, bd, cd; q) (abcd/q; q) π π h(cos 2θ; 1) h(cos θ; a, b, c, d) dθ Observing that the last integrand is even and then equating the right members for the last two equations, we confirm immediately Theorem 1, where we have assumed only abcd/q < 1 for its validity 2 Reversal of the Askey Wilson integral By iterating functional equations, Askey [3, Eq 311] proved another remarkable integral formula, which may be considered as reversal of Theorem 1, in the sense that it almost comes from Theorem 1 by simply inverting the numerators and denominators appearing in the fractions there Theorem 2 (Askey [3, Eq 311]) For qabcd < 1, there holds h(i sinh x; qa, qb, qc, qd) h(cosh 2x; q) dx (q, qab, qac, qad, qbc, qbd, qcd; q) (qabcd; q) log ( q 1)

4 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) Interestingly enough, this theorem can be proved also by utilizing Bailey s identity of 6 ψ 6 -series, which partially justifies the reason why we call the last integral reversal of the Askey Wilson integral Making alternatively the replacements in Eq (1) λ z 2, a z/a, b z/b, c z/c, d z/d; we may reformulate the resulting equation as g(z) (qaz, qa/z, qbz, qb/z, qcz, qc/z, qdz, qd/z; q) (qz 2, q/z 2 ; q) [ ] 6 ψ 1 qz, qz, z/a, z/b, z/c, z/d (q, qab, qac, qad, qbc, qbd, qcd; q) 6 q; qabcd z, z, qaz, qbz, qcz, qdz (qabcd; q) provided that qabcd < 1 for convergence Integrating g(ie x ) over < x <, we can check that Theorem 2 is equivalent to the following integral evaluation, which is strangely independent of the four parameters a, b, c and d Corollary 3 For qabcd < 1, there holds [ 1 iqe 6ψ x, iqe x, ie x /a, ie x /b, ie x /c, ie x ] /d q; 6 ie x, ie x, iqae x, iqbe x, iqce x, iqde x qabcd dx ( log q 1) Proof For simplicity, we prove the integral formula for 0 < q < 1 Then Corollary 3 for the general complex base q with 0 < q < 1 follows from analytic continuation In this case, making the change of variable x y log(q), the integral formula stated in Corollary 3 can equivalently be reformulated as Ω(a, b, c, d) 1 where Ω(a, b, c, d) denotes the integral [ Ω(a, b, c, d) 6ψ 1 iq 1+y, iq 1+y, iq y /a, iq y /b, iq y /c, iq y /d 6 iq y, iq y, iq 1+y a, iq 1+y b, iq 1+y c, iq 1+y d Let W (y) be the last 6 ψ 6 -series [ iq W (y) 6 ψ 1+y, iq 1+y, iq y /a, iq y /b, iq y /c, iq y /d 6 iq y, iq y, iq 1+y a, iq 1+y b, iq 1+y c, iq 1+y d and T n (y) its general summand term ] q; qabcd ] q; qabcd dy T n (y) (iq1+y, iq 1+y, iq y /a, iq y /b, iq y /c, iq y /d; q) n (iq y, iq y, iq 1+y a, iq 1+y b, iq 1+y c, iq 1+y d; q) n (qabcd) n (5) They satisfy, under the replacement of summation index k k + n, therelation W (y) W (y + n) T n (y) where n Z Now for the integral Ω(a, b, c, d), changing the variable y to y + n and then shifting the summation index k k n in the 6 ψ 6 -series, we find the following unusual invariant property Ω(a, b, c, d) Ω n (a, b, c, d) with Ω n (a, b, c, d) T n (y) W (y) dy (4)

5 244 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) With m N being sufficiently large, we may approximate the integrals Ω n (a, b, c, d) by the truncated ones m Ωn m (a, b, c, d) Consider the arithmetic mean m Ω m n T n (y) W (y) dy (a, b, c, d) 1 2m + 1 2m + 1 m Taking into account the limiting relations { m lim m Ωm n (a, b, c, d) Ω n(a, b, c, d) and lim m } T n (y) dy (6) W (y) m T n (y) W (y) it is not difficult to show that Ω(a, b, c, d) 1 by letting m in (6) Infact,thiscanrigorouslybeverifiedasfollowsFirstly,keepinginmindthat qabcd < 1, it is almost trivial to check that the integral Ω(a, b, c, d) converges and the series W (y) is uniformly convergent Observe further that m Ωn m (a, b, c, d) T n (y) m W (y) dy dy W (y + n) m n n dy W (y) Putting these facts together, we assert that for an arbitrarily small ε > 0, there exists a large natural number N N, such that for any m,n N with m n > N, there hold the inequalities m 1 T n (y) W (y) < ε and Ω(a, b, c, d) Ωn m (a, b, c, d) < ε It follows that the right hand side of Eq (6) tends to one as m in view of the following estimation 1 1 m { m T n (y) 2m + 1 W (y) dy 1 m m } 1 + 2m T n (y) W (y) dy < 1 + 2mε 2m + 1 Recall that Ω(a, b, c, d) and Ωn m (a, b, c, d) are bounded integrals, ie, there is a real number M > 0 such that both Ω(a, b, c, d) < M and Ωn m (a, b, c, d) < M Therefore when m, the left hand side of Eq (6) tends to Ω(a, b, c, d), which is justified by the inequalities m Ωn m (a, b, c, d) Ω(a, b, c, d) 2m + 1 m Ω(a, b, c, d) Ωn m (a, b, c, d) 2m + 1 m N 1 + m nm N (2m N)ε < + 2m + 1 Ω(a, b, c, d) Ωn m (a, b, c, d) 2m + 1 Ω(a, b, c, d) Ωn m (a, b, c, d) 2m + 1 2M(N + 1) 2m + 1

6 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) q-beta integral of Karlsson Minton type For the bilateral well-poised series, there is a generalized identity of Karlsson Minton type, which may equivalently be reproduced below Lemma 4 (Chu [7, Theorem 2] and Chu and Wang [9, Corollary 6]) For the n-pairs of complex parameters {x ι, y ι } satisfying the finite conditions n x ι y ι q 1+m ι λ with m ι N 0 and m m ι there holds the following bilateral series identity k 1 q 2k λ 1 λ (a,λ/a, b, d; q) k (qa, qλ/a, qλ/b, qλ/d; q) k ( q 1 ) k n λ bd ( ) m λ (q, q, qλ,q/λ, qλ/ab, qλ/ad, qa/b, qa/d; q) a (qa, q/a, qa/λ, qλ/a, q/b, q/d, qλ/b, qλ/d; q) provided that q 1 λ/ab < 1 for convergence (x ι, y ι ; q) k (qλ/x ι, qλ/y ι ; q) k n (qa/x ι, qa/y ι ; q) m ι (qλ/x ι, qλ/y ι ; q) m ι In view of the finite conditions, the last product can be reformulated as n ( ) m n (qa/x ι, qa/y ι ; q) m ι a (x ι, x ι /λ, qa/y ι, qλ/ay ι ; q) (qλ/x ι, qλ/y ι ; q) m ι λ (x ι /a, ax ι /λ, q/y ι, qλ/y ι ; q) Under the following similar replacements in Lemma 4 λ qz 2, a qz/a, b qz/b, d qz/d, x ι qzu ι, y ι qz/v ι ; the finite conditions will be transformed into u ι /v ι q m ι for 1 ι n and the convergence condition into q bd < 1 Then multiplying across the equation stated in Lemma 4 by (1 z 2 )(1 qz 2 ) n 1 zu ι (1 az)(1 qz/a)(1 bz)(1 dz) 1 zv ι we may express the infinite product-side of the resulting equation as h(z) (q, q, ab, ad, qb/a, qd/a; q) n (av ι, qv ι /a; q) (au ι, qu ι /a; q) (z 2, 1/z 2 ; q) (az, a/z, qz/a, q/az, bz, b/z, dz, d/z) n (u ι z, u ι /z; q) (v ι z, v ι /z; q) For the corresponding bilateral sum-side, splitting it into two partial sums with respect to k 0 and k < 0, and then replacing k by 1 k for the sum with negative indices, we can reformulate the resulting sums as follows:

7 246 W Chu, X Ma / Journal of Combinatorial Theory, Series A 118 (2011) (1 z 2 )(1 qz 2 ) n 1 zu ι h(z) (1 az)(1 qz/a)(1 bz)(1 dz) 1 zv ι k 1 q 1+2k z 2 1 qz 2 (az, qz/a, qz/b, qz/d; q) k (qaz, q 2 z/a, qbz, qdz; q) k ( q bd ) k (1 z 2 )(1 q 1+2k z 2 ) (1 q k az)(1 q 1+k z/a) (qz/b, qz/d; q) k ( ) q k bd (bz, dz; q) k+1 n (qzu ι, qz/v ι ; q) k (qzv ι, qz/u ι ; q) k n (zu ι ; q) k+1 (qz/v ι ; q) k (zv ι ; q) k+1 (qz/u ι ; q) k + (1 z 2 )(1 q 1+2k /z 2 ) (q/bz, q/dz; q) k ( ) n q k (1 q k a/z)(1 q 1+k bd /az) (b/z, d/z; q) k+1 (u ι /z; q) k+1 (q/zv ι ; q) k (v ι /z; q) k+1 (q/zu ι ; q) k Therefore the function h(z) is regular within 0 < z < and can be expanded into the Laurent series at z 0 with the constant term equal to ( ) 2 q k 2 bd 1 q bd Finally, we can integrate h(e iθ ) for π θ π in two different ways according to the two expressions for h(z) Equating the two integral values leads us to the following formula of q-beta integral containing many extra parameters Theorem 5 For the n-pairs of complex parameters {u ι, v ι } subject to the finite conditions n u ι /v ι q m ι with m ι N 0 and m m ι there holds the following integral identity: π h(cos 2θ; 1) h(cos θ; a, q/a, b, d) 0 n h(cos θ; u k ) h(cos θ; v k ) dθ k1 2π/(1 q bd) (q, q, ab, ad, qb/a, qd/aq) n (au ι, qu ι /a; q) (av ι, qv ι /a; q) When m 0, this theorem reduces to the case ac q of the Askey Wilson integral References [1] GE Andrews, R Askey, R Roy, Special Functions, Encyclopedia Math Appl, vol 71, Cambridge University Press, Cambridge, 1999 [2] K Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J Math Anal 18 (1987) [3] R Askey, Beta integrals and q-extensions, in: Proceedings of the Ramanujan Centennial International Conference, Annamalainagar, December 1987, pp [4] R Askey, JA Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem Amer Math Soc 319 (1985) 55 [5] WN Bailey, Series of hypergeometric type which are infinite in both directions, Quart J Math (Oxford) 7 (1936) [6] D Bowman, An easy proof of the Askey Wilson integral and applications of the method, J Math Anal Appl 245 (2) (2000) [7] W Chu, Partial-fraction expansions and well-poised bilateral series, Acta Sci Math (Szeged) 64 (1998) [8] W Chu, Bailey s very well-poised 6 ψ 6 -series identity, J Combin Theory Ser A 113 (6) (2006) [9] W Chu, X Wang, Basic bilateral very well-poised series and Shulkla s 8 ψ 8 -summation formula, J Math Anal Appl 328 (1) (2007) [10] G Gasper, M Rahman, Basic Hypergeometric Series, 2nd edition, Cambridge University Press, Cambridge, 2004 [11] MEH Ismail, D Stanton, On the Askey Wilson integral and Rogers-polynomials, Canad J Math 40 (1988)

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