Dmitry V. Kruchinin

Explicit Formulas for Meixner Polynomials

Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.

Integer properties of a composition of exponential generating functions

In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the results of the paper we get the known properties of the Bell numbers (Touchard’s Congruence for k = 0).In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the results of the paper we get the known properties of the Bell numbers (Touchard’s Congruence for k = 0).

About some properties of polynomials defined by generating functions of form F(t, x)α · G(t, α)x

In this paper, we study the properties of polynomials defined by generating functions of form F(t, x)α · G(t, α)x. We obtain new properties for those polynomials, which allow to obtain interesting identities. As application, using the results of paper we get the identities for the generalized Bernoulli polynomials.

A method for obtaining coefficients of compositional inverse generating functions

The aim of this paper is to show how to obtain expressions for coefficients of compositional inverse generating functions in explicit way. The method is based on the Lagrange inversion theorem and composita of generating functions. Also we give a method of obtaining expressions for coefficients of reciprocal generating functions and consider some examples.