On a specific family of orthogonal polynomials of Bernstein-Szeg\"o type

TL;DR

This paper investigates a special class of orthogonal polynomials related to Bernstein-Szeg"o weights, deriving explicit formulas, quadrature rules, and connections to classical integrals and trigonometric sums.

Contribution

It introduces explicit orthogonal polynomials for a family of weights, enabling new quadrature formulas and finite analogs of classical integrals, linking orthogonal polynomials to elliptic functions.

Findings

Explicit orthogonal polynomials with known roots

Quadrature formulas for these weights

Finite analogs of classical integrals and connections to trigonometric sums

Abstract

We study a class of weight functions on $[-1,1]$, which are special cases of the general weights studied by Bernstein and Szeg\"o, as well as their extentions to the interval $[-a,1]$ for a continuous parameter $a>0$. These weights are parametrized by two positive integers. As these integers tend to infinity, these weights approximate certain weight functions on $\mathbb{R}$ considered in the earlier literature in connection with orthogonal polynomials related to elliptic functions. It turns out that an orthogonal polynomial of certain degree corresponding to these weights has a particularly simple form with known roots. This fact allows us to find explicit quadrature formulas for these weights and construct measures on $\mathbb{R}$ with identical moments. We also find finite analogs of some improper integrals first studied by Glaisher and Ramanujan, and show that some of the functions used in this work are in fact generating functions of certain finite trigonometric power sums.