A note on generating functions and summation formulae for Meixner polynomials of several variables
Abstract The present paper deals with certain generating functions and various elegant summation formulae for Meixner polynomials of several variables.
The Meixner Polynomials in Several Variables
The polynomials are applied in many areas of mathematics, for instance, continued fractions, operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes. On the other hand, generating functions have a great importance in special functions theory. The present study deals with some new properties of the Meixner polynomials in several variables. Firstly, we obtained some results include various families of multilinear and multilateral generating functions for the Meixner polynomials in several variables. In the last section, we get a theorem that gives a bilateral generating functions for these polynomials and the Lauricella functions. Finally, we derive some corollaries of the last theorem.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.