A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator

TL;DR

This paper develops a Taylor expansion theorem for elliptic extensions of the Askey-Wilson operator, enabling new proofs of elliptic hypergeometric identities and computing connection coefficients for elliptic biorthogonal functions.

Contribution

It extends Ismail's expansion for the Askey-Wilson basis to elliptic functions using Rains' operator, providing new proofs and applications in elliptic hypergeometric series.

Findings

Simplified proofs of elliptic Jackson summation and Bailey transformation

Computed connection coefficients for elliptic biorthogonal functions

Extended Jackson's 8-phi-7 summation to nonterminating cases

Abstract

We establish Taylor series expansions in rational (and elliptic) function bases using E. Rains' elliptic extension of the Askey-Wilson divided difference operator. The expansion theorem we consider extends M.E.H. Ismail's expansion for the Askey-Wilson monomial basis. Three immediate applications (essentially already due to Rains) include simple proofs of Frenkel and Turaev's elliptic extensions of Jackson's 8-phi-7 summation and of Bailey's 10-phi-9 transformation, and the computation of the connection coefficients of Spiridonov's elliptic extension of Rahman's biorthogonal rational functions. We adumbrate other examples including the nonterminating extension of Jackson's 8-phi-7 summation and a quadratic expansion.