Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials

TL;DR

This paper derives a series expansion for Askey-Wilson polynomials using advanced hypergeometric series transformations and applies it to explicitly formulate Koornwinder and Macdonald polynomials, confirming Lassalle's conjectures.

Contribution

It introduces a new series expansion for Askey-Wilson polynomials and provides explicit formulas for Koornwinder and Macdonald polynomials, proving Lassalle's conjectures.

Findings

Derived a fourfold series expansion for Askey-Wilson polynomials.

Explicit formulas for Koornwinder polynomials of type BCn.

Confirmed Lassalle's conjectures for Macdonald polynomials with one row diagram.

Abstract

We present a fourfold series expansion representing the Askey-Wilson polynomials. To obtain the result, a sequential use is made of several summation and transformation formulas for the basic hypergeometric series, including the Verma's q-extension of the Field and Wimp expansion, Andrews' terminating q-analogue of Watson's 3F2 sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type BCn (n in Z_>0) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type Bn, Cn and Dn with one row diagram, thereby proving his conjectures.