A partial-sum deformation for a family of orthogonal polynomials

Abstract

There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)=\sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal polynomials $\{p_n(x)\}_{n\ge0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.