A partial-sum deformationfor a family of orthogonal polynomials

Abstract

There are several questions one may ask about polynomials $q_m\left(x\right)=q_m(x;t)={\sum }_{n=0}^m{t}^n{p}_n\left(x\right)$ attached to a family of orthogonal polynomials ${\left\{p_n\left(x\right)\right\}}_{n\ge 0}$. 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 $t$.