An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums

Mathematical Research Letters

Volumn 7, Issue 5-6, 2000, Pages 729-746

  Van Diejen, J F ^a   Spiridonov, V P ^b  

^a UNIVERSITY OF CHILE   (Chile)

^b JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034348688     PISSN: 10732780     EISSN: None     Source Type: Journal    
DOI: 10.4310/mrl.2000.v7.n6.a6     Document Type: Article

References (32)

1. [AW] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985).
2. [A] K. Aomoto, On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), 115-126.
3. [BF] T. H. Baker and P. J. Forrester, Transformation formulas for multivariable basic hypergeometric series, Methods Appl. Anal. 6 (1999), 147-164.
4. [BS] E. P. van den Ban and H. Schlichtkrull, A residue calculus for root systems, Compositio Math. 123 (2000), 27-72.
5. [D] J. F. van Diejen, On certain multiple Bailey, Rogers, and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), 483-508.
6. [EZ] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics 55, Birkhäuser, Boston, 1985.
7. 3, Adv. Math., to appear.
8. [FT] I. B. Frenkel and V. G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand Mathematical Seminars (Eds. V. I. Arnold, I. M. Gelfand, V. S. Retakh, and M. Smirnov), Birkhäuser, Boston, 1997, pp. 171-204.
9. [G1] R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97-105.
10. [G2] _, Some q-beta Integrals on SU(n) and Sp(n) that generalize the Askey-Wilson and Nassrallah-Rahman integrals, SIAM J. Math. Anal. 25 (1994), 441-449.
11. [GR] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications 35, Cambridge University Press, Cambridge, 1990.
12. [HO] G. J. Heckman and E. M. Opdam, Yang's system of particles and Hecke algebras, Ann. of Math. 145 (1997), 139-173, erratum 146 (1997), 749-750.
13. [I] M. Ito, On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), 122-163.
14. n, Mem. Amer. Math. Soc. 108 (1994).
15. [Ma1] I. G. Macdonald, A formal identity for affine root systems, Preprint, 1996.
16. [Ma2] _, Constant term identities, orthogonal polynomials, and affine Hecke algebras, Doc. Math. (1998), DMV Extra Volume ICM I, 303-317.
17. 1 sum. Ramanujan Revisited, (Eds. G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin), Academic Press, Boston, 1988, pp. 473-524.
18. l. Theta Functions Bowdoin 1987, (Eds. L. Ehrenpreis and R.C. Gunning), Proc. Sympos. Pure Math. 49 (part 2), Amer. Math. Soc., Providence, R.I., 1989, pp. 323-359.
19. l Bailey transform and Bailey lemma, Discrete Math. 139 (1995), 319-346.
20. [NR] B. Nassrallah and M. Rahman, Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186-197.
21. 9 rational functions, Canad. J. Math. 38 (1986), 605-618.
22. [R2] _, Biorthogonality of a system of rational functions with respect to a positive measure on [-1,1], SIAM J. Math. Anal. 22 (1991), 1430-1441.
23. [Ru] S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
24. [Sc] M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), 151-169.
25. [Se] J.-P. Serre, A Course in Arithmetic, Springer Verlag, New York, 1973.
26. [S] V. P. Spiridonov, An elliptic beta integral, Proceedings of the Fifth International Conference on Difference Equations and Applications (Temuco, Chile, January 3-7, 2000), (Eds. S. Elaydi, J. Fenner-Lopez, and G. Ladas), Gordon and Breach, to appear.
27. [SZ] V. P. Spiridonov and A. S. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49-83.
28. [St] J. V. Stokman, On BC type basic hypergeometric orthogonal polynomials, Trans. Amer. Math. Soc. 352 (2000), 1527-1579.
29. [Wa] S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Preprint, 2000.
30. [WW] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1986.
31. [W1] J. A. Wilson, Hypergeometric Series, Recurrence Relations and Some New Orthogonal Functions, Ph. D. thesis, University of Wisconsin, Madison, WI, 1978.
32. [W2] _, Orthogonal functions from Gram determinants, SIAM J. Math. Anal. 22 (1991), 1147-1155.