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arXiv · 1508.01794

Orthogonal polynomials related to some Jacobi-type pencils

Sergey M. Zagorodnyuk
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Abstract

In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: $(J_5 - λJ_3) \vec p(λ) = 0$, 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, $\vec p(λ) = (p_0(λ), p_1(λ), p_2(λ),\cdots)^T$, the superscript $T$ means the transposition, with the initial conditions $p_0(λ) = 1$, $p_1(λ) = αλ+ β$, $α> 0$, $β\in\mathbb{R}$. Some orthonormality conditions for the polynomials $\{ p_n(λ) \}_{n=0}^\infty$ are obtained. An explicit example of such polynomials is constructed.

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Sergey M. Zagorodnyuk. 2015-07-28. Orthogonal polynomials related to some Jacobi-type pencils. https://arxiv.org/abs/1508.01794

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