Explicit formulae for generalized Stirling and Eulerian numbers

TL;DR

This paper develops explicit formulas for various generalized Stirling and Eulerian numbers by extending Carlitz's $q$-difference operator, unifying multiple combinatorial number families through a common framework.

Contribution

It introduces a generalized $q$-difference operator to derive explicit sum formulas for extended Stirling and Eulerian numbers, including elliptic and symmetric function variants.

Findings

Derived explicit sum formulas for generalized Stirling numbers.

Extended Carlitz's $q$-Eulerian numbers to Lagrange polynomial form.

Unified multiple number families under a generalized operator framework.

Abstract

In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions, complementary symmetric functions, $r$-Whitney numbers and elliptic analogues of rook, Stirling and Lah numbers. Furthermore, we generalize Carlitz' $q$-Eulerian numbers to a Lagrange polynomial extension. We define them by generalizing Worpitzky's identity appropriately, and derive a recursion and an explicit sum formulae. Special cases include $r$-Whitney Eulerian numbers and elliptic Eulerian numbers.