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+α z2+β z4, 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.
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