On recurrence coefficients of classical orthogonal polynomials

TL;DR

This paper clarifies that existing theorems already cover recent methods for identifying classical orthogonal polynomials via recurrence coefficients, and provides a symbolic algorithm for automated verification, with an application to para-Krawtchouk polynomials.

Contribution

It shows that previous results are encompassed by established theorems and introduces a Mathematica tool for verifying classicality of orthogonal polynomials.

Findings

All recent results are subsumed by two main theorems from 2022.

The symbolic algorithm can verify classicality automatically.

Para-Krawtchouk polynomials are a specific case of classical orthogonal polynomials.

Abstract

In Lett. Math. Phys. 114, 54 (2024) and 115, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is "classical", based solely on its initial recurrence coefficients. This note demonstrates that all the results contained in those works are already encompassed by two general theorems previously established in J. Math. Anal. Appl. 515 (2022), Article 126390. A symbolic algorithm, implemented in Mathematica, is also provided to enable automated verification of the classical character of orthogonal polynomial sequences on quadratic lattices. As an application, it is shown that the so-called para-Krawtchouk polynomials on bi-lattices, discussed in Lett. Math. Phys. 115, 70 (2025), constitute a particular instance of a classical orthogonal family on a linear lattice. Consequently, their algebraic properties follow as a specific case of one of the main theorems established in J. Math. Anal. Appl. 515 (2022), Article 126390.