Open problem in orthogonal polynomials

TL;DR

This paper introduces a new class of orthogonal polynomials arising from quantum mechanics and differential equations, proposing a modification to the Askey scheme, with many properties yet to be analytically derived.

Contribution

It presents a novel class of orthogonal polynomials with continuous and discrete spectra, and proposes a modification to the Askey scheme based on these findings.

Findings

New class of orthogonal polynomials with continuous spectrum

Two discrete versions with finite and countably infinite spectra

Call for further analytical study of their properties

Abstract

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line and two of its discrete versions; one with a finite spectrum and another with countably infinite spectrum. A second class of these new orthogonal polynomials appeared recently while solving a Heun-type equation. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the Askey scheme of hyper-geometric orthogonal polynomials. Up to now, these polynomials are defined by their three-term recursion relations and initial values. However, their other properties like the weight functions, generating functions, orthogonality, Rodrigues-type formulas, etc. are yet to be derived analytically. Due to the prime significance of these polynomials in physics and mathematics, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g., in terms of hypergeometric functions).