Further study on elliptic interpolation formulas for the elliptic Askey-Wilson polynomials and allied identities

TL;DR

This paper introduces elliptic Askey-Wilson polynomials and develops new elliptic interpolation formulas, leading to novel elliptic identities and extensions of classical theta and hypergeometric series results.

Contribution

It establishes general elliptic interpolation formulas using matrix inversion and polynomial representation methods, and characterizes the elliptic Askey-Wilson polynomials as a basis for elliptic interpolation spaces.

Findings

Extended Weierstrass' theta identity

Derived a generalized elliptic Karlsson-Minton identity

Provided an elliptic analogue of Gasper's summation formula

Abstract

In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic interpolation formulas by the methods of matrix inversions and of polynomial representations. Furthermore, we find that the basis of elliptic interpolation space due to Schlosser can be uniquely characterized via the elliptic Askey-Wilson polynomials. As applications of these elliptic interpolation formulas, we establish some new elliptic function identities, including an extension of Weierstrass' theta identity, a generalized elliptic Karlsson-Minton type identity, and an elliptic analogue of Gasper's summation formula for very-well-poised ${}_{6+2m}\phi_{5+2m}$ series.