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arXiv · 2205.15471

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

Gi-Sang Cheon·Tamás Forgács·Arnauld Mesinga Mwafise·Khang Tran
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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.

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Gi-Sang Cheon, Tamás Forgács, Arnauld Mesinga Mwafise, Khang Tran. 2022-05-30. The zero locus and some combinatorial properties of certain exponential Sheffer sequences. https://arxiv.org/abs/2205.15471

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