On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

TL;DR

This paper explores the relationship between two positive measures and their orthogonal polynomial sequences, providing explicit formulas, examples, and identities involving connection coefficients and Radon-Nikodym derivatives.

Contribution

It establishes explicit relationships between orthogonal polynomials associated with measures related by rational functions, including formulas for connection coefficients and Fourier series identities.

Findings

Orthogonal polynomials are related by linear combinations involving previous polynomials.

Explicit formulas for connection coefficients are derived for measures related by rational functions.

Universal identities involving polynomials and connection coefficients are presented.

Abstract

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) is symmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+{\kappa}_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + {\lambda}_{n}b_{n-2}(x) for some sequences {{\kappa}_{n}} and {{\lambda}_{n}}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}} that have a form of Fourier series expansion of the Radon--Nikodym derivative of one measure with respect to the other.