A partial-sum deformation for a family of orthogonal polynomials

TL;DR

This paper explores a partial-sum deformation of orthogonal polynomials, examining the zeros' distribution and structure, revealing natural and elegant properties related to the deformation parameter.

Contribution

It introduces and analyzes a partial-sum deformation of orthogonal polynomials, highlighting its naturalness and structural beauty, especially regarding zeros distribution.

Findings

Zeros' location and distribution depend on the deformation parameter

The deformation reveals elegant structural properties

The partial-sum approach offers new insights into orthogonal polynomial behavior

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 $t$.