$(p, q)$-analogues of the generalized Touchard polynomials and Stirling numbers

TL;DR

This paper introduces $(p, q)$-deformed versions of generalized Touchard polynomials and Stirling numbers, establishing their connections and deriving a recurrence relation that generalizes Spivey's relation.

Contribution

It presents the first $(p, q)$-analogues of these polynomials and numbers, extending previous $q$-deformations to a two-parameter framework with new recurrence relations.

Findings

Established connection between $(p, q)$-deformations and existing $q$-deformations.

Derived a recurrence relation generalizing Spivey's relation for the $(p, q)$-deformed polynomials.

Introduced a post-quantum analog framework for these combinatorial objects.

Abstract

In this paper we introduce a $(p, q)$-deformed analogues of the generalized Touchard polynomials and Stirling numbers, the post-quantum analogues of the $q$-deformed generalized Touchard polynomials and Stirling numbers. The connection between these deformations is established. A recurrence relation for the $(p, q)$-deformed generalized Touchard polynomials is expounded, elucidating a $(p, q)$-deformation of Spivey's relation.