The zero locus and some combinatorial properties of certain exponential Sheffer sequences

TL;DR

This paper investigates the zeros, recurrence relations, and combinatorial properties of a specific Sheffer sequence with an exponential generating function involving quadratic and quartic terms, revealing real or imaginary zeros and connections to generating trees.

Contribution

It introduces a new class of Sheffer sequences with explicit zero properties, recurrence relations, and combinatorial interpretations linked to marked generating trees.

Findings

Zeros are either real or purely imaginary.

Sequence satisfies a four-term recurrence relation.

Coefficients relate to node counts in generating trees.

Abstract

We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form $G(s,z)=e^{czs+\alpha z^{2}+\beta z^{4}}$, where $\alpha, \beta, c \in \mathbb{R}$ with $\beta<0$ and $c\neq 0$. We demonstrate that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a three-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a a given label in certain marked generating trees.