Orthogonal polynomials related to some Jacobi-type pencils

TL;DR

This paper introduces a new class of orthogonal polynomials related to Jacobi-type pencils, exploring their properties, orthonormality conditions, and providing explicit examples, thus extending the theory of orthogonal polynomials.

Contribution

It generalizes orthogonal polynomials on the real line by defining a new relation involving Jacobi matrices and constructs explicit examples and orthonormality conditions.

Findings

Derived orthonormality conditions for the new polynomial class

Constructed explicit examples of these polynomials

Extended the framework of orthogonal polynomials related to matrix pencils

Abstract

In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: $(J_5 - \lambda J_3) \vec p(\lambda) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec p(\lambda) = (p_0(\lambda), p_1(\lambda), p_2(\lambda),\cdots)^T$, the superscript $T$ means the transposition, with the initial conditions $p_0(\lambda) = 1$, $p_1(\lambda) = \alpha \lambda + \beta$, $\alpha > 0$, $\beta\in\mathbb{R}$. Some orthonormality conditions for the polynomials $\{ p_n(\lambda) \}_{n=0}^\infty$ are obtained. An explicit example of such polynomials is constructed.