An identity of Andrews and the Askey-Wilson integral

Transcription

1 Ramanujan J DOI 0.007/s An identity of Andrews and the Askey-Wilson integral Zhi-Guo Liu Received: 6 July 2007 / Accepted: 7 January 2008 Springer Science+Business Media, LLC 2008 Abstract Andrews (Adv. Math. 4:37 72, 98) derived a four-variable q-series identity, which is an extension of the Ramanujan ψ summation. In this paper, we shall give a simple evaluation of the Askey-Wilson integral by using this identity. Keywords q-series Andrews identity Askey-Wilson integral Ramanujan ψ summation Mathematics Subject Classification (2000) Primary 05A30 33D5 33D05 33D45 33D60 Let 0 <q<, the q-shifted factorials are defined by and n (a; q) 0, (a; q) n ( aq k ), n 0,, 2,... (a; q) lim n (a; q) n ( aq k ). Andrews [] proved a remarkable q-series identity, which is an extension of Ramanujan s ψ summation. The Andrews identity also includes Ramanujan s reciprocity The author was supported by the National Science Foundation of China, PCSIRT and Innovation Program of Shanghai Municipal Education Commission. Z.-G. Liu ( ) Department of Mathematics, East China Normal University, Shanghai , People s Republic of China zgliu@math.ecnu.edu.cn

2 Z.-G. Liu theorem for a certain q-series [3, 8]. The following is an equivalent form of the Andrews identity [0]. Theorem If bu <, bv <, then (q/bu, acuv; q) n v (bv) n u (av, cv; q) n+ (q/bv, acuv; q) n (bu) n (au, cu; q) n+ v(q,qv/u,u/v,abuv,acuv,bcuv; q) (au, av, bu, bv, cu, cv; q). () Askey and Wilson [2] used contour integration and a clever elliptic function argument to evaluate an important q-beta integral, which is now known as the Askey- Wilson integral. Theorem 2 (Askey-Wilson) If max a, b, c, d <, then where and 0 h(cos θ; a,b,c,d) dθ 2π(abcd; q), (2) (q,ab,ac,ad,bc,bd,cd; q) h(cos θ; a) (ae iθ,ae iθ ; q) h(cos θ; a,a 2,...,a m ) h(cos θ; a )h(cos θ; a 2 ) h(cos θ; a m ). In this paper we will use () to give a very simple proof of (2). This proof does not require the orthogonality relation for the q-hermite polynomials. For other proofs, see [4 7, 9, ]. Proof It is obvious that () can be written as (q,v/u,u/v,abuv,acuv,bcuv; q) (au, av, bu, bv, cu, cv; q) (q/bu, acuv; q) n ( v/u) (bv) n (av, cv; q) n+ + ( u/v) Putting u e iθ and v e iθ yields h(cos θ; a,b,c) ( e 2iθ (qe iθ /b, ac; q) n ) (ae iθ,ce iθ (be iθ ) n ; q) n+ (q/bv, acuv; q) n (au, cu; q) n+ (bu) n. (3)

3 An identity of Andrews and the Askey-Wilson integral + ( e 2iθ ) (qe iθ /b, ac; q) n (ae iθ,ce iθ ; q) n+ (be iθ ) n. (4) Inspecting the first series in the above equation we see that in this series, every term contributes the positive powers of e iθ, except the n 0 term. Hence the constant term of the Fourier expansion of this series can only occur in the n 0 term, which is ( e 2iθ ) ( ae iθ )( ce iθ ). It is obvious that the constant term is. So we have ( e 2iθ ) Replacing θ by θ gives ( e 2iθ ) Substituting (5) and (6)into(4) gives (qe iθ /b, ac; q) n (ae iθ,ce iθ ; q) n+ (be iθ ) n + a n e inθ. (5) n (qe iθ /b, ac; q) n (ae iθ,ce iθ ; q) n+ (be iθ ) n + a n e inθ. (6) h(cos θ; a,b,c) 2 + a n (e inθ + e inθ ). (7) n Integrating the above equation over [,π] and using the fact e inθ dθ 2πδ n,0 in the resulting equation, we immediately have dθ 4π, h(cos θ; a,b,c) or h(cos θ; a,b,c) dθ 4π. (8) To complete the proof of the Askey-Wilson integral, we also need the q-leibniz rule for the q-differential operator n [ ] n Dq n f(a)g(a) q k(k n) Dq k k f(a)dn k q g(aq k ), (9) where the q-differential operator D q is defined by n D q f(a) (f (a) f (aq)), a Dn q f D q(d n q f). (0)

4 Z.-G. Liu It is understood that throughout this paper, D q acts on the variable a.theq-binomial coefficient is defined as [ ] n. () k (q; q) k k Now we apply Dq n to both sides of (8) to obtain h(cos θ; b,c) Dn q By the q-leibniz rule, we have D n q h(cos θ,a) dθ 4π Dq n (q, bc; q). (2) A n(a,b,c), (3) where A n (a,b,c) n [ ] n (ac; q) k b k c n k. (4) k Putting b e iθ and c e iθ gives D n q h(cos θ,a) Combining (2), (3), and (5) yields A n(a, e iθ,e iθ ). (5) h(cos θ; a) A n (a, e iθ,e iθ ) dθ 4πA n(a,b,c). (6) h(cos θ; a,b,c) Multiplying both sides by d n / and then summing on n, 0 n,we have h(cos θ; a,b,c) 4π Using the q-binomial theorem, we find that A n (a,b,c)d n d n A n (a, e iθ,e iθ ) dθ (7) (ac; q) k (bd) k (q; q) k A n (a,b,c)d n. (cd) n k k nk (abcd; q) (bd, cd; q). (8)

5 An identity of Andrews and the Askey-Wilson integral It follows that d n A n (a, e iθ,e iθ ) Combining (7) (9) we conclude that (ad; q) h(cos θ; d). (9) h(cos θ; a,b,c,d) dθ 4π(abcd; q). (20) (q,ab,ac,ad,bc,bd,cd; q) The integrand is an even function of θ and so (20) implies (2). This completes the proof of Theorem 2. The author would like to thank the editor for many useful comments and sugges- Acknowledgement tions. References. Andrews, G.E.: Ramanujan s lost notebook. I. Partial θ-functions. Adv. Math. 4, (98) 2. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54, 39 (985) 3. Berndt, B.C., Chan, S.H., Yeap, B.P., Yee, A.J.: A reciprocity theorem for certain q-series found in Ramanujan s lost notebook. Ramanujan J. 3, (2007) 4. Chen, W.Y.C., Liu, Z.-G.: Parameter augmentation for basic hypergeometric series, II. J. Comb. Theory Ser. A 80, (997) 5. Ismail, M.E.H., Stanton, D.: On the Askey-Wilson and Rogers polynomials. Can. J. Math. 40, (988) 6. Ismail, M.E.H., Stanton, D., Viennot, G.: The combinatorics of q-hermite polynomials and the Askey- Wilson integral. Eur. J. Comb. 8, (987) 7. Kalnins, E.G., Miller, W.: Symmetry techniques for q-series: Askey-Wilson polynomials. Rocky Mt. J. Math. 9, (989) 8. Kang, S.-Y.: Generalizations of Ramanujan s reciprocity theorem and their applications. J. Lond. Math. Soc. 75, 8 34 (2007) 9. Liu, Z.-G.: q-hermite polynomials and a q-beta integral. Northeast. Math. J. 3, (997) 0. Liu, Z.-G.: Some operator identities and q-series transformation formulas. Discrete Math. 265, 9 39 (2003). Rahman, M.: A simple evaluation of Askey and Wilson s integral. Proc. Am. Math. Soc. 92, (984)