A generalization of Selberg’s beta integral

Gustafson, Robert A.

doi:10.1090/S0273-0979-1990-15852-5

A generalization of Selberg's beta integral

Author: Robert A. Gustafson
Journal: Bull. Amer. Math. Soc. 22 (1990), 97-105
MSC (1985): Primary 33A15, 33A75, 05A19
DOI: https://doi.org/10.1090/S0273-0979-1990-15852-5
MathSciNet review: 1001607
Full-text PDF Free Access

1. K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987), no. 2, 545–549. MR 876291, https://doi.org/10.1137/0518042
2. George E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 191–224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR 0399528
3. Richard Askey, Beta integrals and 𝑞-extensions, Proceedings of the Ramanujan Centennial International Conference (Annamalainagar, 1987) RMS Publ., vol. 1, Ramanujan Math. Soc., Annamalainagar, 1988, pp. 85–102. MR 993347
4. Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, https://doi.org/10.1090/memo/0319
5. E. W. Barnes, A new development of the theory of hypergeometric functions, Proc. London, Math. Soc. (2) 6 (1908), 141-177.
6. Louis de Branges, Tensor product spaces, J. Math. Anal. Appl. 38 (1972), 109–148. MR 417337, https://doi.org/10.1016/0022-247X(72)90122-9
7. Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140–156. MR 143556, https://doi.org/10.1063/1.1703773
8. F. Garvan, A proof of the Macdonald-Morris root system conjecture for F, preprint.
9. F. Garvan and G. H. Gonnet, A proof of the two parameter q-case of the Macdonald-Morris root system conjecture for S(F, preprint.
10. J. Gunson, Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Mathematical Phys. 3 (1962), 752–753. MR 148401, https://doi.org/10.1063/1.1724277
11. Robert A. Gustafson, Some 𝑞-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), no. 1, 69–119. MR 1139492, https://doi.org/10.1090/S0002-9947-1994-1139492-3
12. R. Gustafson, Some multidimensional beta type integrals, preprint.
13. Laurent Habsieger, La 𝑞-conjecture de Macdonald-Morris pour 𝐺₂, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 6, 211–213 (French, with English summary). MR 860819
14. Kevin W. J. Kadell, A proof of the 𝑞-Macdonald-Morris conjecture for 𝐵𝐶_{𝑛}, Mem. Amer. Math. Soc. 108 (1994), no. 516, vi+80. MR 1140650, https://doi.org/10.1090/memo/0516
15. I. G. Macdonald, Affine root systems and Dedekind’s 𝜂-function, Invent. Math. 15 (1972), 91–143. MR 357528, https://doi.org/10.1007/BF01418931
16. I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161–174. MR 322069, https://doi.org/10.1007/BF01431421
17. I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988–1007. MR 674768, https://doi.org/10.1137/0513070
18. M. L. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, New York-London, 1967. MR 0220494
19. W. G. Morris, Constant term identities for finite and affine root systems: conjectures and theorems, Ph. D. thesis, Univ. of Wisconsin-Madison, 1982.
20. E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152, https://doi.org/10.1007/BF01388841
21. Mizan Rahman, Another conjectured 𝑞-Selberg integral, SIAM J. Math. Anal. 17 (1986), no. 5, 1267–1279. MR 853529, https://doi.org/10.1137/0517088
22. S. Ramanujan, A class of definite integrals, Quart. J. Math. 48 (1920), 294-310.
23. Atle Selberg, Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71–78 (Norwegian). MR 18287
24. James A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), no. 4, 690–701. MR 579561, https://doi.org/10.1137/0511064
25. Kenneth G. Wilson, Proof of a conjecture by Dyson, J. Mathematical Phys. 3 (1962), 1040–1043. MR 144627, https://doi.org/10.1063/1.1724291
26. Doron Zeilberger, A proof of the 𝐺₂ case of Macdonald’s root system-Dyson conjecture, SIAM J. Math. Anal. 18 (1987), no. 3, 880–883. MR 883574, https://doi.org/10.1137/0518065
27. Doron Zeilberger, A unified approach to Macdonald’s root-system conjectures, SIAM J. Math. Anal. 19 (1988), no. 4, 987–1013. MR 946656, https://doi.org/10.1137/0519068
28. Doron Zeilberger and David M. Bressoud, A proof of Andrews’ 𝑞-Dyson conjecture, Discrete Math. 54 (1985), no. 2, 201–224. MR 791661, https://doi.org/10.1016/0012-365X(85)90081-0