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

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 $c\ne 0$. 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.