[Paper Review] Additions to the formula lists in "Hypergeometric orthogonal polynomials and their $q$-analogues" by Koekoek, Lesky and Swarttouw

Tom H. Koornwinder|arXiv (Cornell University)|Jan 4, 2014

Mathematical functions and polynomials60 references22 citations

TL;DR

This report supplements the comprehensive formula lists in Koekoek, Lesky, and Swarttouw's "Hypergeometric orthogonal polynomials and their $q$-analogues" by compiling additional formulas for (q-)hypergeometric orthogonal polynomials that were omitted in the original work. The author provides new identities, transformations, and generating functions—particularly for $q$-Laguerre, Stieltjes-Wigert, and discrete $q$-Hermite polynomials—supported by references or proofs, enhancing the utility of the canonical reference for researchers in special functions and orthogonal polynomials.

ABSTRACT

This paper gives a rather arbitrary choice of formulas for ($q$-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book "Hypergeometric orthogonal polynomials and their $q$-analogues" by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed here, in particular for the numbering of subsections and of references.

Motivation & Objective

• To compile and systematize additional formulas for (q-)hypergeometric orthogonal polynomials not included in the main reference by Koekoek, Lesky, and Swarttouw.
• To address gaps in standard formula lists, particularly outside core categories like orthogonality and recurrence relations.
• To provide accessible, referenced derivations or proofs for formulas frequently needed in research on special functions.
• To extend the utility of the canonical text by including formulas for $q$-Laguerre, Stieltjes-Wigert, and discrete $q$-Hermite polynomials.
• To support researchers by offering a curated, reference-annotated collection of formulas that are either missing or hard to locate in standard sources.

Proposed method

• Systematically collects formulas from standard references such as [GR], [DLMF], [AAR], and [Sz], focusing on (q-)hypergeometric orthogonal polynomials.
• Applies known transformations, such as quadratic and $q$-hypergeometric identities, to derive new relations (e.g., between $q$-Laguerre and discrete $q$-Hermite polynomials).
• Uses generating functions—particularly the $q$-exponential generating function (169)—to derive expansion formulas like (170) for $x^n$ in terms of $p_n(x;a;q)$.
• Corrects and re-expresses weight functions, such as for Stieltjes-Wigert polynomials, by adjusting parameters to match known forms (e.g., (179) for $w(x)$).
• Establishes connections between different polynomial families via quadratic transformations, such as relating $p_n(x^2; q^{-1}; q^2)$ to $h_{2n}(x;q)$ in (171).
• Derives new expansion formulas using generating function coefficients, as in (170), by equating powers of $t$ in the expansion of $x^n$.

Experimental results

Research questions

• RQ1 What additional formulas for (q-)hypergeometric orthogonal polynomials are missing from the canonical reference by Koekoek, Lesky, and Swarttouw?
• RQ2 How can generating functions be used to derive explicit expansions of monomials $x^n$ in terms of orthogonal polynomial bases?
• RQ3 What are the correct weight functions and normalization constants for Stieltjes-Wigert and $q$-Laguerre polynomials, especially in alternative parametrizations?
• RQ4 How do quadratic transformations link $q$-Laguerre and discrete $q$-Hermite polynomials, and what are the precise coefficient identities?
• RQ5 What is the correct parametrization of the weight function for Stieltjes-Wigert polynomials to match known results in the literature?

Key findings

• The generating function identity (169) provides a $q$-exponential generating function for the little $q$-Laguerre polynomials, enabling the derivation of expansion formulas.
• Formula (170) gives a precise expansion of $x^n$ as a finite sum over little $q$-Laguerre polynomials $p_k(x;a;q)$, with coefficients involving $q$-Pochhammer symbols.
• The quadratic transformation (171) relates $p_n(x^2; q^{-1}; q^2)$ to the discrete $q$-Hermite I polynomial $h_{2n}(x;q)$, with a coefficient involving $(q; q^2)_n$.
• Formula (177) establishes a similar link between $q$-Laguerre polynomials with $eta = -1/2$ and discrete $q$-Hermite II polynomials $ ilde{h}_{2n}(x;q)$, confirming consistency with earlier results.
• The corrected weight function (179) for Stieltjes-Wigert polynomials is expressed in terms of $rac{ u}{ u}$, with $ u^2 = -1/(2 ext{ln} q)$, aligning with known parametrizations in [DLMF] and [Sz].
• The normalization constant $h_0$ for $q$-Laguerre polynomials is given by (174), with a continuous limit for $eta o ext{integer}$, and explicitly evaluated as $h_n = q^{-rac{1}{2}eta(eta+1)}(q;q)_eta ext{log}(q^{-1})$ for $eta ext{ integer} o 0$ (175).