An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

TL;DR

This paper introduces two new infinite sequences of orthogonal polynomials, called $X_1$-Jacobi and $X_1$-Laguerre, which start with degree one and are characterized by a Sturm-Liouville problem, expanding classical polynomial systems.

Contribution

The paper defines new orthogonal polynomial sequences that begin with degree one and characterizes them via Sturm-Liouville problems, extending classical orthogonal polynomial theory.

Findings

$X_1$-Jacobi and $X_1$-Laguerre polynomials are orthogonal and form bases in their Hilbert spaces.

A converse theorem similar to Bochner's theorem is proved for these new polynomial systems.

Rodrigues-type formulas are derived for both polynomial sequences.

Abstract

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\infty)$, respectively, and they are a basis of the corresponding $L^2$ Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second order operator has a complete set of polynomial eigenfunctions $\{p_i\}_{i=1}^\infty$, then it must be either the $X_1$-Jacobi or the $X_1$-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the $X_1$ polynomial sequences.