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+αz 2 +βz 4 , 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 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. MSC: 05A15,