Zeros of linear combinations of orthogonal polynomials

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.