Polynomial solutions of differential-difference equations

TL;DR

This paper studies the zeros of polynomial solutions to a class of differential-difference equations, focusing on conditions for real, simple, and interlacing zeros, encompassing classical orthogonal and special polynomials.

Contribution

It provides general conditions for the zeros' properties in polynomial solutions to differential-difference equations, extending to classical and special polynomial families.

Findings

Zeros are real and simple under certain conditions.

Zeros of consecutive polynomials are interlacing.

Results include classical orthogonal, Euler-Frobenius, Bell, and other polynomials.

Abstract

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlacing. Our result holds for general classes of polynomials but includes sequences of classical orthogonal polynomials as well as Euler-Frobenius, Bell and other polynomials.