Modular hypergeometric residue sums of elliptic Selberg integrals

Letters in Mathematical Physics

Volumn 58, Issue 3, 2001, Pages 223-238

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

^a UNIVERSIDAD DE TALCA   (Chile)

^b JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

Author keywords

Hypergeometric sums; Jacobi forms; Residue calculus; Selberg integrals

Indexed keywords

EID: 0001700934     PISSN: 03779017     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1014567500292     Document Type: Article

References (36)

1. Aomoto, K.: On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebra Comb. 8 (1998), 115-126.
2. Askey, R. and Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), No. 319.
3. Baker, T. H. and Forrester, P. J.: Transformation formulas for multivariate basic hypergeometric series, Methods Appl. Anal. 6 (1999), 147-164.
4. Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Exactly solvable SOS models: local height probabilities and theta function identities, Nuclear Phys. B 290 (1987), 231-273.
5. Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Exactly solvable SOS models, II: Proof of the star-triangle relation and combinatorial identities, In: M. Jimbo, T. Miwa and A. Tsuchiya (eds). Conformal Field Theory and Lattice Models, Adv. Stud. Pure Math. 16, Kinokuniya, Tokyo, 1988, pp. 17-122.
6. Denis R. Y. and Gustafson, R. A.: An SU(n) q/-beta integral transformation and multiple hypergeometric series identities, SIAM J. Math. Anal. 23 (1992), 552-561.
7. Van Diejen, J. F.: On certain multiple Bailey, Rogers, and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), 483-508.
8. Van Diejen, J. F. and Spiridonov, V. P.: An elliptic Macdonald-Morris conjecture and modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729-746.
9. Van Diejen, J. F. and Spiridonov, V. P.: Elliptic Selberg integrals, Internal. Math. Res. Notices 2001(20), 1083-1110.
10. Eichler, M. and Zagier, D.: The Theory of Jacobi Forms, Progr. in Math. 55, Birkhäuser, Boston, 1985.
11. 3, Adv. Math. 156 (2000), 44-76.
12. Frenkel, I. B. and Turaev, V. G. Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, In: V. I. Arnold. I. M. Gelfand, V. S. Retakh and M. Smirnov (eds), The Arnold-Gelfand Mathematical Seminars, Birkhäuser, Boston, 1997, pp. 171-204.
13. Gustafson, R. A.: The Macdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple Lie algebras, In: N. K. Thakare, K. C. Sharma, T. T. Raghunathan (eds), Ramanujan International Symposium on Analysis, Macmillan, New Dehli, 1989, pp. 185-224.
14. Gustafson R. A.: Some q-beta and Mellin-Barnes integrals with many parameters associated to classical groups, SIAM J. Math. Anal. 23 (1992), 525-224.
15. Gasper, G. and Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications 35, Cambridge Univ. Press. Cambridge, 1990.
16. Itô M.: On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), 122-163.
17. Itô, M.: Symmetry classification for Jackson integrals associated with irreducible root systems, Compositio Math. 129 (2001), 325-340.
18. Kajihara, Y.: Euler transformation formula for multiple basic hypergeometric series of type A and some applications, Preprint 2001.
19. 1 summation theorem for Macdonald polynomials, Ramanujan J. 2 (1998), 379-386.
20. Krattenthaler, C.: The major counting of nonintersecting lattice paths and generating functions for tableaux. Mem., Amer. Math. Soc. 115 (1995), No. 552.
21. Macdonald, I. G.: A formal identity for affine root systems, Preprint, 1996.
22. l, In: L. Ehrenpreis and R. C. Gunning (eds), Theta Functions Bowdoin 1987, Proc. Symp. Pure Math. 49 (part 2), Amer. Math. Soc., Providence, R.I. 1989, 323-359.
23. l Bailey transform and Bailey lemma, Constr. Approx. 9 (1993), 473-500.
24. 1 Bailey transtorm and (Bailey lemma, Discrete Math. 139 (1995), 319-346.
25. Nassrallah, B. and Rahman, M.: Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186-197.
26. Opdam, E. M.: An analogue of the Gauss summation formula for hypergeometric functions related to root systems, Math. Z. 212 (1993), 313-336.
27. Rahman M.: Biorthoaonality of a system of rational functions with respect to a positive measure on [-1, 1], SIAM J. Math. Anal. 22 (1991) 1430-1441.
28. Rosengren, H.: A proof of a multivariable elliptic summation formula conjectured by Warnaar, Contemp. Math., to be published.
29. Ruijsenaars, S. N. M.: First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
30. Schlosser, M.: Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), 151-169.
31. Serre, J.-P.: A Course in Arithmetic, Springer, New York, 1973.
32. Spiridonov, V. P.: On an elliptic beta function, Russian Math. Surveys 56(1) (2001), 185-186.
33. Spiridonov, V. P. and Zhedanov, A.: Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49-83.
34. Spiridonov, V. P. and Zhedanov, A.: Classical biorthogonal rational functions on elliptic grids, C.R. Math. Acad. Sci. Soc. R. Can 22 (2000), 70-76.
35. Warnaar, S. O.: Summation and transformation formulas for elliptic hypergeometric series, Preprint, 2000.
36. Whittaker, E. T. and Watson, G. N.: A Course of Modern Analysis, Cambridge University Press, Cambridge, 1986.