On certain multiple Bailey, Rogers and Dougall type summation formulas

Abstract

A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6{\psi }_6$ summation formula and its Dougall type ${}_5{H}_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers ${}_6{\varphi }_5$ and Dougall ${}_5{F}_4$ sums (both nonterminating and terminating). The terminating sums may be used to arrive at product formulas for the norms of recently introduced ($q$-)Racah polynomials in several variables.

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Cited by in corpus (16)

• Summation and transformation formulas for elliptic hypergeometric series
• A simple proof of Bailey's very-well-poised 6-psi-6 summation
• Elementary derivations of identities for bilateral basic hypergeometric series
• A new multivariable 6-psi-6 summation formula
• Orthogonality of Macdonald polynomials with unitary parameters
• A proof of a multivariable elliptic summation formula conjectured by Warnaar
• Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems
• A generalization of the Sears--Slater transformation and elliptic Lagrange interpolation of type $B{C}_n$
• Branching Formula for Macdonald-Koornwinder Polynomials
• BC_n-symmetric polynomials
• Reduction formulae for Karlsson-Minton type hypergeometric functions
• Elliptic hypergeometric series on root systems
• Generalized Cherednik-Macdonald identities
• Difference Fourier transforms for nonreduced root systems
• Proof of a summation formula for an ${\tilde{A}}_n$ basic hypergeometric series conjectured by Warnaar
• Affine Hall-Littlewood functions for $A_1^{(1)}$ and some constant term identities of Cherednik-Macdonald-Mehta type