Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group

Dongkyu Lim,Feng Qi

doi:10.3390/math6120332

Abstract

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results.

Highlights

The Stirling numbers arise in a variety of analytic and combinatorial problems
The Stirling numbers of second kind S(n, k ) are the numbers of ways to partition a set of n elements into k nonempty subsets
The purpose of this paper is to investigate some interesting symmetric identities involving the Stirling polynomials Sn ( x ) under the finite symmetric group Sn

Summary

Introduction

The Stirling numbers arise in a variety of analytic and combinatorial problems. They were introduced in the eighteenth century by James Stirling. Some combinatorial identities for the Stirling numbers of these two kinds are studied and collected in [1,2,3,4,5,6,7,8] and closely related references. The Stirling numbers of second kind S(n, k ) are the numbers of ways to partition a set of n elements into k nonempty subsets. The symmetric identities of some special polynomials, such as higher order Bernoulli polynomials Bn , higher order q-Euler polynomials, degenerate generalized. We can obtain some new symmetric identities involving the Stirling polynomials Sn ( x )

Symmetric Identities of Stirling Polynomials

Symmetric Identities via Higher Order Bernoulli Polynomials