Abstract
In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.
Highlights
This paper is a study in q-calculus
Received: 10 June 2020; Accepted: 14 July 2020; Published: 3 August 2020. In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy
The work contained in this paper is directly connected to properties of the Askey–Wilson polynomials pn ( x; a|q) ([1] §14.1) which are at the very top of the q-Askey scheme (see e.g., ([1] Chapter 14))
Summary
This paper is a study in q-calculus (typically taken with |q| < 1). The q-calculus (introduced by such luminaries as Leonhard Euler, Eduard Heine and Garl Gustav Jacobi) is a calculus of finite differences which becomes the standard infinitesimal calculus (introduced by Isaac Newton and Gottfried Wilhelm Leibniz) in the limit as q → 1. (which gives restrictions on the values of the free parameters), the q-inverse Askey–Wilson polynomials represent a finite-family of basic hypergeometric orthogonal polynomials The main focus of this survey paper will be to exhaustively describe the transformation identities for the terminating basic hypergeometric functions which appear as representations for these polynomials. Some of these transformation identities are well-known in the literature, but we give the transformation identities for these basic hypergeometric functions which are obtained by the symmetry of the polynomials under parameter interchange, and under the map θ 7→ −θ, for x = cos θ. See [4,5,6]
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