A class of orthogonal functions given by a three term recurrence formula

TL;DR

This paper introduces a new class of orthogonal functions with a three-term recurrence relation, extending symmetric orthogonal polynomials and linking to orthogonal polynomials on the unit circle, with applications in quadrature rules.

Contribution

It presents a novel class of orthogonal functions satisfying a specific orthogonality and recurrence relation, extending existing polynomial classes and connecting to unit circle polynomials.

Findings

New class of orthogonal functions with three-term recurrence

Connection to orthogonal polynomials on the unit circle

Development of quadrature rules based on zeros

Abstract

The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on $[-1,1]$, has a complete connection to the orthogonal polynomials on the unit circle. Quadrature rules and other properties based on the zeros of these functions are also considered.