Keywords: Daehee numbers, Hypergeometric Daehee numbers and polynomials
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
pp 5962.
DOI: 10.12691/tjant1112
Received October 07, 2013; Revised November 20, 2013; Accepted December 04, 2013
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
As is known, the Daehee polynomials are defined by the generating function to be
 (1.1) 
(see ^{[5, 6, 7, 9, 10, 11, 12]}).
In the special case, are called the Daehee numbers.
Let and denote the rings of padic integers, the fields of padic numbers and the completion of algebraic closure of The padic norm is normalized by Let be the space of uniformly differentiable functions on For the padic invariant integral on is defined by
 (1.2) 
(see ^{[7, 8]}).
Let be the translation of with Then, by (1.2), we get
 (1.3) 
As is known, the Stirling number of the first kind is defined by
 (1.4) 
and the Stirling number of the second kind is given by the generating function to be
 (1.5) 
(see ^{[2, 3, 4]}).
For the Bernoulli polynomials of order are defined by the generating function to be
 (1.6) 
(see ^{[1, 2, 9]}).
When are called the Bernoulli numbers of order
A hypergeometric series is a series for which and the ratio of consecutive terms is a rational function of the summation index i.e., one for which
with and polynomials. In this case, is called a hypergeometric term. The functions generated by hypergeometric series are called generalized hypergeometric functions. If the polynomials are completely factored, the ratio of successive terms can be written
 (1.7) 
(see ^{[13]}),
where the factor of in the denominator is present for historical reasons of notation, and the resulting generalized hypergeometric function is written
 (1.8) 
(see ^{[13]}).
If and the function becomes a traditional hypergeometric function Many sums can be written as generalized hypergeometric functions by inspections of the ratios of consecutive terms in the generating hypergeometric series.
We introduce the hypergeometric Daehee numbers and polynomials. From our definition, we can derive some interesting properties related to the hypergeometric Daehee numbers and polynomials.
2. The HyperGeometric Daehee Numbers and Polynomials
First, we consider the following integral representation associated with falling factorial sequences:
 (2.1) 
By (2.1), we get
 (2.2) 
(see ^{[6]}), where with
For with let us take Then, from (1.3), we have
 (2.3) 
By (1.1) and (2.3), we see that
 (2.4) 
(see ^{[6]}).
Therefore, by (2.4), we obtain the following Lemma.
Lemma 1. For we have
For it is known that
 (2.5) 
(see ^{[4, 5, 6]}).
Thus, by (2.5), we get
 (2.6) 
where are the Bernoulli polynomials of order
In the special case, are called the nth Bernoulli numbers of order
From (2.4), we note that
 (2.7) 
(see ^{[6]}).
Thus, by (2.7), we get
 (2.8) 
and, from (2.5), we have
 (2.9) 
(see ^{[6]}).
Therefore, by (2.8) and (2.9), we obtain the following Lemma.
Lemma 2. For we have
and
By Lemma 1, we easily see that
 (2.10) 
(see ^{[6]}), where are the ordinary Bernoulli numbers.
From Lemma 2, we have
 (2.11) 
(see ^{[6]}), where are the Bernoulli polynomials defined by generating function to be
Therefore, by (2.10) and (2.11), we obtain the following corollary.
Corollary 3. For we have
In (2.4), we have
 (2.12) 
and
 (2.13) 
(see ^{[6]}).
Therefore, by (2.12) and (2.13), we obtain the following Lemma.
Lemma 4. For we have
In particular,
Remark. For by (2.11), we have
(see ^{[6]}).
Now, we define the hypergeometric Daehee polynomials
 (2.14) 
where
For example, we have
 (2.15) 
Thus the hypergeometric Daehee number are defined by
 (2.16) 
Note that is the Daehee number.
 (2.17) 
where
 (2.18) 
Therefore, by (2.18), we obtain the following theorem.
Theorem 5. For we have
In (2.17), we have
 (2.19) 
Therefore, by (2.19), we obtain the following theorem.
Theorem 6. For we have
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