Orthogonal polynomials related to some Jacobi-type 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.

arXiv comment9 pages

Cited by in corpus (1)

• The Inverse Spectral Problem for Jacobi-Type Pencils