Some Basic Extensions of Gustafson-Rakha's Multivariate Basic Hypergeometric Series

Medhat A. Rakha

Mathematics Department, Faculty of Science, Suez Canal University, Ismailia 41522,
Egypt

marakha@hotmail.com

Annals of Combinatorics 6 (1) p.103-115 March, 2002

Abstract:

In this paper we extend some special cases of the multivariate basic hypergeometric
series associated to the roots system of type
that has been established and proved in [1]. For both types of the series, we will
prove that when *m=2n, n=1* one of the series is equivalent to Jackson's
sum, while the other series is equivalent to the basic Gauss' sum.

References:

1. R. Askey and J. Wilson, A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 13 (1982) 651每655.

2. R.Y. Denis and R.A. Gustafson, An SU(n) q-beta integral transformation and multiple hypergeometric series identities, SIAM J. Math. Anal. 23 (2) (1992) 552每561.

3. F.J. Dyson, Three identities in combinatory analysis, J. London Math. Soc. 18 (1943) 35每39.

4. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990.

5. R.A. Gustafson, Multilateral summation theorems for ordinary and basic hypergeometric series in U(n), SIAM J. Math. Anal. 18 (6) (1987) 1576每1596.

6. R.A. Gustafson, A generalization of Selberg＊s beta integral, Bull. Amer. Math. Soc. 22 (1) (1990) 97每105.

7. R.A. Gustafson, Some q-beta integrals on SU(n) and Sp(n) which generalize the Askey- Wilson and Nasrallah-Rahman integrals, SIAM J. Math. Anal. 25 (1994) 441每449.

8. R.A. Gustafson and M.A. Rakha, q-Beta integrals and multivariate basic hypergeometric series associated to root systems of type Am, Ann. Combin. 4 (2000) 347每373.

9. W.J. Holman III, Summation theorems for hypergeometric series in U(n), SIAM J. Math. Anal. 11 (1980) 523每532.

10. W.J. Holman III, L.C. Biedenharn, and J.D. Louck, On hypergeometric series well-poised in SU(n), SIAM J. Math. Anal. 17 (1976) 529每541.

11. I.G. Macdonald, Orthogonal polynomials associated with root systems, S谷m. Lothar. Combin. B45a (2000) 40 pp.

12. S.C. Milne, Hypergeometric series well-poised in SU(n), and a generalization of Biedenharn＊s G-functions, Adv. Math. 36 (1980) 169每211.

13. S.C. Milne, A q-analog of hypergeometric series well-poised in SU(n) and invariant Gfunctions, Adv. Math. 58 (1) (1985) 1每60.

14. S.C. Milne, Basic hypergeometric series very well-poised inU(n), SIAM J. Math. Anal. 122 (1) (1987) 223每256.

15. S.C. Milne, A q-analog of the Gauss summations theorem for hypergeometric series inU(n), Adv. Math. 72 (1) (1988) 59每131.

16. W.G. Morris, Constant term identities for finite and affine root systems: conjectures and theorems, Ph.D. Thesis, Univ. of Wisconsin-Madson, 1982.

17. M.A. Rakha, On the root systems of the calssical & exceptional Lie algebras, Mathematica Programs, Adv. Mod. & Anal. 35 No. 1 (2) (1999) 1每12.