# 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 _{3}*F*_{2}
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*and

_{n}*D*with one row diagram, thereby proving his conjectures.

_{n}2010 Math. Subj. Class. 33D45, 33D52.

**Keywords:**Askey–Wilson polynomial.

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