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Annals of Combinatorics 4
(2000) 347-373
*q*-Beta Integrals and Multivariate Basic Hypergeometric
Series Associated to Root Systems of Type *A*_{m}
Robert A. Gustafson and Medhat
A. Rakha
Department of Mathematics,
Texas A & M University, College Station, Texas 77843, USA
robert.gustafson@math.tamu.edu
Department of Mathematics,
Faculty of Science, Suez Canal University, Ismailia - 41522, Egypt
mrakha@ismailia.ie-eg.com
Received September 28, 1998
**AMS Subject Classification**: 33A15,
33A30, 33A65, 33A75
**Abstract.** In this paper we establish
and prove two new *q*-beta integrals and two multivariate basic hypergeometric
series associated to the root system of A_{m}. We give a generalization
of Jackson's well-poised
sum for one kind of series and a generalization of the *q*-Gauss sum
for the other kind of series.
**Keywords**: *q*-series, *q*-beta
integrals, integral transformations, hypergeometric series well poised
on Lie algebras
**References**
1. R. Askey and J. Wilson, *A set of hypergeometric orthogonal polynomials*, SIAM J. Math.
Anal. 13 (1982) 651–655.
2. R. Askey and J.Wilson, *Some basic hypergeometric orthogonal polynomials that generalize
Jacobi polynomials*, Mem. AMS 319 (1985) 55.
3. W.N. Bailey, *Generalized hypergeometric series*, Cambridge Math. Tract No. 32, Cambridge
University Press, Cambridge, 1935; Hafner, New York, 1964, reprint.
4. R.Y. Denis and R.A. Gustafson, *An SU*(*n*) *q-beta integral transformation and multiple hypergeometric
series identities*, SIAM J. Math. Anal. 23 (1992) 552–561.
5. F.J. Dyson,* Three identities in combinatory analysis*, J. London Math. Soc. 18 (1943) 35–39.
6. G. Gasper and M. Rahman, *Basic hypergeometric series*, Cambridge University Press, Cambridge,
1990.
7. R.A. Gustafson, *Multilateral summation theorems for ordinary and basic hypergeometric
series*, SIAM J. Math. Anal. 18 (1987) 1576–1596.
8. R.A. Gustafson, *A generalization of Selberg’s beta integral*, Bulletin of AMS 22 (1990)
97–105.
9. 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) 441–449.
10. W.J. Holman III, *Summation theorems for hypergeometric series in U*(*n*), SIAM J. Math.
Anal. 11 (1980) 523–532.
11. 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) 529–541.
12. I.G. Macdonald, *Orthogonal polynomials associated with root systems*, preprint.
13. S. C. Milne, *A q-analog of hypergeometric series well-poised in SU*(*n*) *and invariant Gfunctions*,
Vol. 58, No. 1, October (1985), 1–60.
14. S.C. Milne, *Basic hypergeometric series very well-poised in U*(*n*), SIAM J. of Math. Anal.
122 (1987) 223–256.
15. S.C. Milne, *A q-analog of the Gauss summations theorem for hypergeometric series inU*(*n*).
Adv. Math. 72 (1988) 59–131.
16. S.C. Milne, *Hypergeometric series well-poised in SU*(*n*), *and a generalization of Biedenharn’s
G-functions*, Adv. Math. 36 (1980) 169–211.
17. S.C. Milne, *Balanced *_{3}ø_{2} summation theorems for U(*n*) *basic hypergeometric series*, Adv.
Math. 131 (1997) 93–187.
18. W.G. Morris, *Constant term identities for finite and affine root systems: conjectures and
theorems*, Ph.D. Thesis, University of Wisconsin-Madson, 1982.
19. L.J. Slater, *Generalized hypergeometric functions*, Cambridge University Press, London,
New York, 1966.
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