Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric 8Φ7 sum of Milne-Gustafson type and the (one-dimensional) modular hypergeometric 8ε7 sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.

[1]  A 1Ψ1 Summation Theorem for Macdonald Polynomials , 1998 .

[2]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[3]  Mizan Rahman,et al.  Projection Formulas, a Reproducing Kernel and a Generating Function for q-Wilson Polynomials , 1985 .

[4]  Simon Ruijsenaars,et al.  First order analytic difference equations and integrable quantum systems , 1997 .

[5]  Jean-Pierre Serre.  A Course in Arithmetic , 1973 .

[6]  Masahiko Ito.  Symmetry Classification for Jackson Integrals Associated with Irreducible Reduced Root Systems , 2001, Compositio Mathematica.

[7]  Michio Jimbo,et al.  Exactly solvable SOS models: local height probabilities and theta function identities , 1987 .

[8]  Stephen C. Milne,et al.  Consequences of the Al and Cl Bailey transform and Bailey lemma , 1995, Discret. Math..

[9]  R. A. Gustafson.  Some q-beta and Mellin-Barnes integrals with many parameters associated to the classical groups , 1992 .

[10]  Transformation formulae for multivariable basic hypergeometric series , 1998, math/9803146.

[11]  J. F. van Diejen.  On certain multiple Bailey, Rogers and Dougall type summation formulas , 1997, math/9712265.

[12]  Hjalmar Rosengren.  A proof of a multivariable elliptic summation formula conjectured by Warnaar , 2001 .

[13]  Michio Jimbo,et al.  Exactly Solvable SOS Models II: Proof of the star-triangle relation and combinatorial identities , 1988 .

[14]  E. Opdam.  An analogue of the Gauss summation formula for hypergeometric functions related to root systems , 1993 .

[15]  Alexei Zhedanov,et al.  Spectral Transformation Chains and Some New Biorthogonal Rational Functions , 2000 .

[16]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[17]  J. F. van Diejen,et al.  Elliptic Selberg integrals , 2001 .

[18]  Alexander Varchenko,et al.  The Elliptic Gamma Function and SL(3, Z)⋉Z3 , 2000 .

[19]  K. Aomoto.  On Elliptic Product Formulas for Jackson Integrals Associated with Reduced Root Systems , 1998 .

[20]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[21]  J. F. van Diejen,et al.  An Elliptic Macdonald-Morris Conjecture and Multiple Modular Hypergeometric Sums , 2000 .

[22]  S. Warnaar,et al.  Summation and transformation formulas for elliptic hypergeometric series , 2000, math/0001006.

[23]  S. Milne,et al.  TheCℓ Bailey transform and Bailey lemma , 1993 .

[24]  Mizan Rahman.  Biorthogonality of a system of rational functions with respect to a positive measure on [-1,1] , 1991 .

[25]  Vyacheslav P. Spiridonov,et al.  On the elliptic beta function , 2001 .

[26]  Vladimir Turaev,et al.  Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions , 1997 .

[27]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[28]  Michael J. Schlosser.  Summation theorems for multidimensional basic hypergeometric series by determinant evaluations , 2000, Discret. Math..

[29]  Masahiko Ito.  On a Theta Product Formula for Jackson Integrals Associated with Root Systems of Rank Two , 1997 .

[30]  Christian Krattenthaler,et al.  The major counting of nonintersecting lattice paths and generating functions for tableaux , 1995 .

[31]  R. A. Gustafson,et al.  An SU(n) q -beta integral transformation and multiple hypergeometric series identities , 1992 .