When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

TL;DR

This paper investigates when linear combinations of orthogonal polynomials produce new orthogonal sequences, providing conditions and characterizations, especially for the case where the combination length is three.

Contribution

It offers necessary and sufficient conditions for the orthogonality of linear combinations of orthogonal polynomials with fixed length, and characterizes families where these combinations are also orthogonal.

Findings

Conditions for orthogonality of linear combinations derived

Jacobi matrix interpretation provided

Characterization for the case k=2

Abstract

Given $\{P_n \}$ a sequence of monic orthogonal polynomials, we analyze their linear combinations $\{Q_n \}$with constant coefficients and fixed length $k+1$. Necessary and sufficient conditions are given for the orthogonality of the monic sequence $\{Q_n \}$ as well as an interesting interpretation in terms of the Jacobi matrices associated with $\{P_n \}$ and $\{Q_n \}$. Moreover, in the case $k=2$, we characterize the families $\{P_n \}$ such that the corresponding polynomials $\{Q_n \}$ are also orthogonal.