# 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 BCn (n ∈ 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.

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