Touchard like polynomials and generalized Stirling numbers

TL;DR

This paper generalizes Touchard polynomials through exponential operators and operational formalism, establishing new properties and connections to Stirling numbers, thus broadening the theoretical framework of these combinatorial polynomials.

Contribution

It introduces a novel method using exponential operators to extend Touchard polynomials and explores their relationship with Stirling numbers.

Findings

Derived new properties of generalized Touchard polynomials

Established connections to various forms of Stirling numbers

Provided a unified operational framework for these polynomials

Abstract

The theory of Touchard polynomials is generalized using a method based on the definition of exponential operators, which extend the notion of the shift operator. The proposed technique, along with the use of the relevant operational formalism, allows the straightforward derivation of properties of this family of polynomials and their relationship to different forms of Stirling numbers.