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This article is cited in 2 scientific papers (total in 2 papers)
Some transformation formulas associated with Askey–Wilson polynomials and Lassalle's formulas for Macdonald–Koornwinder polynomials
A. Hoshinoa, M. Noumib, J. Shiraishic a 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.
Received: June 6, 2014; in revised form December 27, 2014
Citation:
A. Hoshino, M. Noumi, J. Shiraishi, “Some transformation formulas associated with Askey–Wilson polynomials and Lassalle's formulas for Macdonald–Koornwinder polynomials”, Mosc. Math. J., 15:2 (2015), 293–318
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https://www.mathnet.ru/eng/mmj560 https://www.mathnet.ru/eng/mmj/v15/i2/p293
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