Zeros of linear combinations of orthogonal polynomials

TL;DR

This paper investigates the real zeros of finite linear combinations of consecutive orthogonal polynomials, proving the existence of sequences where these zeros are real and simple for sufficiently large degrees.

Contribution

It establishes the existence of orthogonal polynomial sequences with linear combinations having only real, simple zeros beyond a certain degree.

Findings

Existence of orthogonal polynomial sequences with real, simple zeros for linear combinations.

Zeros are real and simple for all sufficiently large n.

Results hold for any positive measure on the real line.

Abstract

Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_jp_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $\mu$ there always exists a sequence of orthogonal polynomials with respect to $\mu$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $\gamma_j$'s.