In this paper, we present a unified grammatical interpretation of the numbers that satisfy kind four-term recurrence relation, including Bell triangle, coefficients modified Hermite polynomials, and Bessel polynomials. Additionally, as an application, criterion for real zeros row-generating polynomials is also presented.

We prove a formula for the linearization coefficients of the general Sheffer polynomials, which unifies all the special known results for Hermite, Charlier, Laguerre, Meixner and MeixnerPollaczek polynomials. Furthermore, we give a new and explicit real version of the corresponding formula for Meixner-Pollaczek polynomials. Our proof is based on some explicit bijections and sign-reversing weigh...

As a generalization of 2D Bernoulli polynomials, neo-Bernoulli polynomials are introduced from a point of view involving the use of nonexponential generating functions. Their relevant recurrence relations, the differential equations satisfied by them and some other properties are obtained. Especially, we obtain the relationships between them and neo-Hermite polynomials. We also study some other...