Theta Functions, Elliptic Hypergeometric Series, and Kawanaka's Macdonald Polynomial Conjecture

TL;DR

This paper introduces a new theta-function identity that proves Kawanaka's Macdonald polynomial conjecture and provides a novel transformation formula for multivariable elliptic hypergeometric series, expanding the mathematical toolkit.

Contribution

The paper presents a new theta-function identity and applies it to prove a longstanding conjecture, also deriving a new transformation formula for elliptic hypergeometric series.

Findings

Proved Kawanaka's Macdonald polynomial conjecture

Derived a new transformation formula for elliptic hypergeometric series

Established a novel theta-function identity

Abstract

We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.