This article is cited in 2 papers

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

A. Hoshino^a, M. Noumi^b, J. Shiraishi<sup>c

a</sup> Kagawa National College of Technology, 355 Chokushi-cho, Takamatsu, Kagawa 761-8058, Japan
^b Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
^c Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

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 $_3F_2$ sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type $BC_n$ ($n\in\mathbb Z_{>0}$) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type $B_n$, $C_n$ and $D_n$ with one row diagram, thereby proving his conjectures.

Key words and phrases: Askey–Wilson polynomial.

MSC: 33D45, 33D52

Received: June 6, 2014; in revised form December 27, 2014

Language: English

DOI: 10.17323/1609-4514-2015-15-2-293-318

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