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 $$q_n(x)=\sum _{j=0}^K{\gamma }_j{p}_{n-j}(x),n\ge K,$$ where ${\gamma }_j$, $j=0,\cdots ,K$, are real numbers with ${\gamma }_0=1$, ${\gamma }_K\ne 0$ (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.