Orthogonal polynomials related to some Jacobi-type pencils
Sergey M. Zagorodnyuk

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: , where is a Jacobi matrix and is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, , the superscript means the transposition, with the initial conditions , , , . Some orthonormality conditions for the polynomials are obtained. An explicit example of such polynomials is constructed.
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Taxonomy
TopicsMatrix Theory and Algorithms · Mathematical functions and polynomials · Molecular spectroscopy and chirality
