Difference equations related to Jacobi-type pencils

TL;DR

This paper explores difference equations linked to Jacobi-type pencils, analyzing their solutions, spectral properties, and related orthogonal polynomials satisfying fourth-order differential equations.

Contribution

It introduces new difference equations associated with Jacobi-type pencils and constructs classical orthogonal polynomials satisfying fourth-order differential equations.

Findings

Constructed the basic set of solutions for the 4th order difference equation.

Investigated spectral properties of the truncated pencil.

Derived special matrix orthogonality relations.

Abstract

In this paper we study various difference equations related to Jacobi-type pencils. By a Jacobi-type pencil one means the following pencil: $J_5 - \lambda J_3$, 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. The basic set of solutions for the corresponding $4$-th order difference equation is constructed. Spectral properties of the truncated pencil and some special matrix orthogonality relations are investigated. Classical type orthogonal polynomials satisfying a $4$-th order differential equation are constructed.