A Note on Type 2 w-Daehee Polynomials

Minyoung Ma,Dongkyu Lim

doi:10.3390/math7080697

Abstract

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

Highlights

Motivated by Equation (1), we present type 2 w-Daehee polynomials via p-adic invariant integral on Z p as follows: Z
For the case of w = 1, w = 12 and w = 14, the symmetry of the type 2 w-Daehee polynomials are related to the works of the type 2 Daehee polynomials, those of well-known Daehee polynomials [1], and we can modify relate to those of the Catalan Daehee polynomials in [9], respectively
These are motivated from the pursuit of the symmetric properties of the type 2 Daehee polynomials and numbers, which are defined and investigated by Kims [11]

Summary

Introduction

Let p be a fixed prime number. Throughout this paper, Z p , Q p and C p will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p , respectively. The p-adic norm | · | p is normalized as | p| p = 1/p.It is common knowledge that the usual Bernoulli numbers Bn are given by the generating function to be, for t ∈ C p , ∞ t tn = B n∑ n! , et − 1 n =0 which can be written symbolically as eBt = t/(et − 1), interpreted to mean that Bn must be replaced byBn . In addition, usual Bernoulli polynomials Bn ( x ) are defined by, for x ∈ C p , n Bn ( x ) = ∑ l Bl x n − l . l =0

Objectives

Discussion

Conclusion