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.