On the zeros of certain Sheffer sequences and their cognate sequences

TL;DR

This paper investigates the zeros of Sheffer and cognate polynomial sequences, revealing their distribution patterns and limiting behaviors, especially for Appell and broader Sheffer families.

Contribution

It introduces the concept of cognate sequences for Sheffer polynomials and analyzes their zero distributions and limiting probability measures.

Findings

Zeros are real or on a line for large n in certain sequences

Zero distributions converge to specific probability measures

Identifies conditions for zero location in Sheffer sequences

Abstract

Given a Sheffer sequence of polynomials, we introduce the notion of an associated sequence called the cognate sequence. We study the relationship between the zeros of this pair of associated sequences and show that in case of an Appell sequence, as well as a more general family of Sheffer sequences, the zeros of the members of each sequence (for large n) are either real, or lie on a line $\Re (z)=c$. In addition to finding the zero locus, we also find the limiting probability distribution function of such sequences.