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)=eczs+αz2+βz4, where α,β,c∈R with β<0 and c≠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 four-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 given label in certain marked generating trees.