Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Contiguity relations for finite families of orthogonal polynomials in the Askey scheme

Nicolas Crampé, Lucia Morey, Luc Vinet, Meri Zaimi

TL;DR

This work provides a complete classification of contiguity relations for finite families of orthogonal polynomials in the ($q$-)Askey scheme and connects these relations to spectral transforms. It establishes necessary and sufficient constraints for contiguity types $A_2$, $B_2$, and $B'_2$, and solves them across twelve discrete families, including $q$-Racah, Hahn, Krawtchouk, and Bannai–Ito polynomials. A key finding is that all $A_2$ contiguity relations are Christoffel or Geronimus transforms, with $B_2$/$B'_2$ relations derivable from the $A_2$ cases (notably for $q$-Racah). The paper further extends these ideas to Bannai–Ito polynomials via $q\to -1$ limits, develops a generalization to ${}_4\phi_3$ functions, and situates the results within a broader spectral-transform framework with potential applications to kernel polynomials and discrete integrable structures.

Abstract

This paper classifies the contiguity relations for finite families of polynomials within the ($q$-)Askey scheme. The necessary and sufficient conditions for the existence of these contiguity relations are presented first. These conditions are then solved, yielding a comprehensive list of contiguity relations for these various families of polynomials. Furthermore, we demonstrate that all contiguity relations correspond to spectral transforms.

Contiguity relations for finite families of orthogonal polynomials in the Askey scheme

TL;DR

This work provides a complete classification of contiguity relations for finite families of orthogonal polynomials in the (-)Askey scheme and connects these relations to spectral transforms. It establishes necessary and sufficient constraints for contiguity types , , and , and solves them across twelve discrete families, including -Racah, Hahn, Krawtchouk, and Bannai–Ito polynomials. A key finding is that all contiguity relations are Christoffel or Geronimus transforms, with / relations derivable from the cases (notably for -Racah). The paper further extends these ideas to Bannai–Ito polynomials via limits, develops a generalization to functions, and situates the results within a broader spectral-transform framework with potential applications to kernel polynomials and discrete integrable structures.

Abstract

This paper classifies the contiguity relations for finite families of polynomials within the (-)Askey scheme. The necessary and sufficient conditions for the existence of these contiguity relations are presented first. These conditions are then solved, yielding a comprehensive list of contiguity relations for these various families of polynomials. Furthermore, we demonstrate that all contiguity relations correspond to spectral transforms.
Paper Structure (26 sections, 7 theorems, 119 equations)

This paper contains 26 sections, 7 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Contiguity relations
  3. Contiguity relations of type $A_2$
  4. Contiguity relations of type $B_2$
  5. Contiguity relations of type $B'_2$
  6. Spectral transforms and orthogonal polynomials
  7. Solutions of the constraints $(\mathfrak{C}_{A_2})$, $(\mathfrak{C}_{B_2})$ and $(\mathfrak{C}_{B'_2})$
  8. --Racah
  9. --Hahn
  10. Dual --Hahn
  11. Quantum --Krawtchouk
  12. --Krawtchouk
  13. Affine --Krawtchouk
  14. Dual --Krawtchouk
  15. Racah
  16. ...and 11 more sections

Key Result

Proposition 2.2

The polynomials $R_{i}(x;{\boldsymbol{\rho}})$ chosen such that $\lambda_{x,{\boldsymbol{\rho}}}=\zeta\lambda_{{\overline{x}},\overline{{\boldsymbol{\rho}}}}-\xi$ satisfy the contiguity relation eq:cons1 of type $A_2$ (with the convention $\Phi_{0}^{-1,+} = 0$ and with $\Phi_{0}^{0,+} \neq 0$) if and only if the coefficients are given (up to a global normalization) by and the constraints $(\mathf

Theorems & Definitions (18)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 8 more