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Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$

Tom H. Koornwinder

TL;DR

This work completes a uniform $q=1$ realization of the Verde-Star parametrization for the Askey scheme, expressing almost all non-Hermite families through four essential parameters via the data $h_k$, $x_k$, and $g_k$. It provides explicit $q=1$ data, a hypergeometric (and dual) representation for the resulting polynomials, and a scheme based on degree triples that mirrors the classical Askey framework while clarifying limit transitions from the $q$-case. The paper shows that, apart from Hermite polynomials, the entire $q$-Askey landscape collapses to a four-parameter Verde-Star description, with a uniform parametrization and explicit data for each family. It also details how $q o1$ limits connect the $q$-Verde-Star scheme to the Verde-Star scheme, offering concrete mappings and illustrating the nonuniformity of a single parametrization across $q$-levels. The results deepen understanding of the structural relationships among orthogonal polynomial families and their limit transitions, and provide practical hypergeometric expressions for a broad class of $q=1$ polynomials.

Abstract

Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way.

Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$

TL;DR

This work completes a uniform realization of the Verde-Star parametrization for the Askey scheme, expressing almost all non-Hermite families through four essential parameters via the data , , and . It provides explicit data, a hypergeometric (and dual) representation for the resulting polynomials, and a scheme based on degree triples that mirrors the classical Askey framework while clarifying limit transitions from the -case. The paper shows that, apart from Hermite polynomials, the entire -Askey landscape collapses to a four-parameter Verde-Star description, with a uniform parametrization and explicit data for each family. It also details how limits connect the -Verde-Star scheme to the Verde-Star scheme, offering concrete mappings and illustrating the nonuniformity of a single parametrization across -levels. The results deepen understanding of the structural relationships among orthogonal polynomial families and their limit transitions, and provide practical hypergeometric expressions for a broad class of polynomials.

Abstract

Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the Askey scheme together with their hypergeometric representations by three sequences of polynomials in , two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the -case to the case , although this cannot be done in a uniform way.
Paper Structure (18 sections, 1 theorem, 38 equations, 2 figures)

This paper contains 18 sections, 1 theorem, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Note
  3. Verde-Star's theorem
  4. The general set-up and two examples
  5. Verde-Star's theorem and some symmetries
  6. $x\leftrightarrow h$ duality and ($q$-)hypergeometric form
  7. The general $q$-case
  8. The $q=1$ case
  9. The Verde-Star scheme
  10. Scheme with degree triples
  11. Scheme with uniform parameters
  12. Limit from the $q$-Verde-Star scheme to the Verde-Star scheme
  13. A list of $q\to1$ limits
  14. Explicit data for the families in Figure \ref{['21']}
  15. Some worked-out $q\to1$ limits
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let the sequences $h_k,x_k,g_k$ satisfy 75, 76, 77. Then $u_n(x)$ satisfies a three-term recurrence relation if and only if

Figures (2)

  • Figure 1: The Verde-Star scheme using degree triples
  • Figure 2: The Verde-Star scheme with indication of vanishing parameters

Theorems & Definitions (8)

  • Example 2.1: Askey--Wilson polynomials
  • Example 2.2: Wilson polynomials
  • Theorem 2.3: Verde-Star 1
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Example 4.1: Askey--Wilson $\to$ Wilson