Abstract
We consider the Witt-type formula for the n th twisted Daehee numbers and polynomials and investigate some properties of those numbers and polynomials. In particular, the n th twisted Daehee numbers are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind.
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1 Introduction
In this paper, we assume that , and will, respectively, denote the rings of p-adic integers, the fields of p-adic numbers and the completion of algebraic closure of . The p-adic norm is normalized by . Let be the space of uniformly differentiable functions on . For , the p-adic invariant integral on is defined by
Let be the translation of f with . Then, by (1), we get
As is known, the Stirling number of the first kind is defined by
and the Stirling number of the second kind is given by the generating function to be
For , the Bernoulli polynomials of order α are defined by the generating function to be
When , are called the Bernoulli numbers of order α.
For , let be the p-adic locally constant space defined by
where is the cyclic group of order . It is well known that the twisted Bernoulli polynomials are defined as
and the twisted Bernoulli numbers are defined as .
Recently, Kim and Kim introduced the Daehee numbers and polynomials which are given by the generating function to be
In the special case, , are called the n th Daehee numbers.
In the viewpoint of generalization of the Daehee numbers and polynomials, we consider the n th twisted Daehee polynomials defined by the generating function to be
In the special case, , are called the n th twisted Daehee numbers.
In this paper, we give a p-adic integral representation of the n th twisted Daehee numbers and polynomials, which are called the Witt-type formula for the n th twisted Daehee numbers and polynomials. We can derive some interesting properties related to the n th twisted Daehee numbers and polynomials. For this idea, we are indebted to papers [9, 10].
2 Witt-type formula for the n th twisted Daehee numbers and polynomials
First, we consider the following integral representation associated with falling factorial sequences:
By (8), we get
where with .
For with , let us take . Then, from (2), we have
By (9) and (10), we see that
Therefore, by (11), we obtain the following theorem.
Theorem 1 For , we have
For , it is known that
Thus, replacing t by in (12), we get
where are the Bernoulli polynomials of order n.
In the special case, , are called the n th Bernoulli numbers of order n.
From (11), we note that
Thus, by (14), we get
and, from (12), we have
Therefore, by (15) and (16), we obtain the following theorem.
Theorem 2 For , we have
and
By Theorem 1, we easily see that
where are the ordinary Bernoulli numbers.
From Theorem 2, we have
where are the Bernoulli polynomials defined by a generating function to be
Therefore, by (17) and (18), we obtain the following corollary.
Corollary 3 For , we have
In (11), we have
Replacing t by , we put
Therefore, we have
where is the Stirling number of the second kind.
Hence,
Therefore, we have
In particular,
Therefore, by (20) and (23), we obtain the following theorem.
Theorem 4 For , we have
In particular,
Remark For , by (18), we have
For , the rising factorial sequence is defined by
Let us define the n th twisted Daehee numbers of the second kind as follows:
By (26), we get
From (26) and (27), we have
Therefore, by (28), we obtain the following theorem.
Theorem 5 For , we have
Let us consider the generating function of the n th twisted Daehee numbers of the second kind as follows:
From (2), we can derive the following equation:
where .
By (29) and (30), we get
Let us consider the n th twisted Daehee polynomials of the second kind as follows:
Then, by (32), we get
From (33), we get
Therefore, by (34), we obtain the following theorem.
Theorem 6 For , we have
From (32) and (33), we have
Replacing t by , we get
Therefore, we have
Hence,
Therefore, we have
Therefore, by (37) and (38), we obtain the following theorem.
Theorem 7 For , we have
From Theorem 1 and (26), we have
and
Therefore, by (40) and (41), we obtain the following theorem.
Theorem 8 For , we have
and
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The authors are grateful for the valuable comments and suggestions of the referees.
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Park, JW., Rim, SH. & Kwon, J. The twisted Daehee numbers and polynomials. Adv Differ Equ 2014, 1 (2014). https://doi.org/10.1186/1687-1847-2014-1
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DOI: https://doi.org/10.1186/1687-1847-2014-1