Study of the $p,q$-deformed Touchard polynomials

TL;DR

This paper introduces a two-parameter deformation of Touchard polynomials using Tsallis' q-exponential, providing explicit formulas for new statistics on set partitions and enabling broader function analysis.

Contribution

It presents a novel two-parameter deformation of Touchard polynomials based on Tsallis' q-exponential, with explicit formulas for associated statistics and coefficients.

Findings

Derived explicit formulas for deformed Touchard polynomials

Established new statistics on set partitions

Enabled evaluation of Taylor series coefficients for various functions

Abstract

A two-parameter deformation of the Touchard polynomials, based on the NEXT $q$-exponential function of Tsallis, defines two statistics on set partitions. The generating function of classical Touchard polynomials is a composition of two exponential functions. By applying analysis of a combinatorial structure of the deformed exponential function, we establish explicit formulae for both statistics. Moreover, the explicit formulae for the deformed Touchard polynomials makes possible to evaluate coefficients of Taylor series expansion for wide variety of functions with different values of parameters $p$ and $q$.