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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,

This paper establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays. In particular, the case of Riordan arrays derived from Jacobi elliptic functions which are parameterized by the elliptic modulus k will be treated here. Some concrete examples of such Riordan arrays are presented via a recursive formula. Furthermore, some applications of these Riordan arrays corresponding to the solutions of the PDE modelling nonlinear wave phenomena for shallow water waves and nonlinear electric line transmission are highlighted.

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