Moscow Mathematical Journal

Volume 15, Issue 2, April–June 2015 pp. 293–318.

Some Transformation Formulas Associated with Askey–Wilson Polynomials and Lassalle’s Formulas for Macdonald–Koornwinder Polynomials

Authors: A. Hoshino (1), M. Noumi (2), and J. Shiraishi (3)
Author institution:(1) Kagawa National College of Technology, 355 Chokushi-cho, Takamatsu, Kagawa 761-8058, Japan
(2) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
(3) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Summary:

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 BC_n (n ∈ 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.

2010 Math. Subj. Class. 33D45, 33D52.

Keywords: Askey–Wilson polynomial.

Contents Full-Text PDF