On combinatorial properties and the zero distribution of certain Sheffer sequences

TL;DR

This paper investigates the combinatorial structures and zero distribution of a specific Sheffer sequence generated by quadratic polynomials, revealing that most zeros lie on a vertical line in the complex plane.

Contribution

It introduces new combinatorial interpretations and analyzes the zero distribution of Sheffer sequences with quadratic generating functions using Riordan matrices.

Findings

Zeros of high-degree polynomials mostly lie on the line Re(x)=1/2

Provides combinatorial interpretations of the Sheffer sequence coefficients

Identifies two exceptional zeros outside the main zero distribution line

Abstract

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.