Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

TL;DR

This paper introduces new generating functions for generalized Stirling, array, and Eulerian polynomials and numbers, deriving identities and applications through functional, differential equations, and integral transforms.

Contribution

It constructs novel generating functions for these generalized numbers and polynomials, providing new identities and applications using advanced mathematical tools.

Findings

Derived new functional and differential equations for the polynomials and numbers.

Established multiplication formulas and recurrence relations.

Applied p-adic Volkenborn integral and Laplace transform to obtain identities.

Abstract

The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized {\lambda}-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.