A research on the new polynomials involved with the central factorial numbers, Stirling numbers and others polynomials

Jung Yoog Kang,Cheon Seoung Ryoo

doi:10.1186/1029-242x-2014-26

Abstract

Recently, many mathematicians have studied different kinds of the Euler, Bernoulli, and Genocchi polynomials. In this paper, we give another definition of the polynomials . We find some theorems and identities related to polynomials containing the central factorial numbers and Stirling numbers. We also derive interesting relations between the polynomials and the Euler polynomials and the Genocchi polynomials.

Highlights

The Stirling numbers of the first kind s(n, k) are defined by [ ]n s(n, k)xk = x(x – )(x – ) · · · (x – n + ). ( . ) k=The generating function of ( . ) is as follows: log( + x) k = k! ∞ xn s(n, k) . n! n=kFrom ( . ) and ( . ), we become aware of some properties of the Stirling numbers of the first kind, s(n, k) [ ]: s(n, k) = s(n, k – ) – (n – )s(n, k), with
From ( . ) and ( . ), we become aware of some properties of the Stirling numbers of the first kind, s(n, k) [ ]: s(n, k) = s(n, k – ) – (n – )s(n, k), with s(n, ) = (n ∈ N), s(n, n) = n ∈ Z+ = N ∪, s(n, ) = (– )n– (n – )! (n ∈ N), s(n, k) = (k > n or k < )
In Section, we derive some special relations of the polynomials Un(α)(x) and the Euler polynomials

Summary

Introduction

We usually define the central factorial numbers T(n, k) by the following expansion formula [ , ]: n The Euler numbers En and Euler polynomials En(x) are defined by For a real or complex parameter α, the generalized Euler polynomials of degree n are defined by the following generating functions: We consider the addition theorem for these polynomials.

Results

Conclusion