The Hyper-Geometric Daehee Numbers and Polynomials
1Department of Mathematics Education, Kyungpook National University, Taegu, Republic of Korea
2Department of Mathematics, Kyungpook National University, Taegu, Republic of Korea
We consider the hyper-geometric Daehee numbrers and polynomials and investigate some properties of those numbers and polynomials.
Keywords: Daehee numbers, Hyper-geometric Daehee numbers and polynomials
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
Cite this article:
- Park, Jin-Woo, Seog-Hoon Rim, and Jongkyum Kwon. "The Hyper-Geometric Daehee Numbers and Polynomials." Turkish Journal of Analysis and Number Theory 1.1 (2013): 59-62.
- Park, J. , Rim, S. , & Kwon, J. (2013). The Hyper-Geometric Daehee Numbers and Polynomials. Turkish Journal of Analysis and Number Theory, 1(1), 59-62.
- Park, Jin-Woo, Seog-Hoon Rim, and Jongkyum Kwon. "The Hyper-Geometric Daehee Numbers and Polynomials." Turkish Journal of Analysis and Number Theory 1, no. 1 (2013): 59-62.
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As is known, the Daehee polynomials are defined by the generating function to be
In the special case, are called the Daehee numbers.
Let and 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 with Then, by (1.2), 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 are defined by the generating function to be
When are called the Bernoulli numbers of order
A hyper-geometric 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 hyper-geometric term. The functions generated by hyper-geometric series are called generalized hyper-geometric functions. If the polynomials are completely factored, the ratio of successive terms can be written
where the factor of in the denominator is present for historical reasons of notation, and the resulting generalized hyper-geometric function is written
If and the function becomes a traditional hyper-geometric function Many sums can be written as generalized hyper-geometric functions by inspections of the ratios of consecutive terms in the generating hyper-geometric series.
We introduce the hyper-geometric Daehee numbers and polynomials. From our definition, we can derive some interesting properties related to the hyper-geometric Daehee numbers and polynomials.
2. The Hyper-Geometric Daehee Numbers and Polynomials
First, we consider the following integral representation associated with falling factorial sequences:
By (2.1), we get
(see ), where with
For with let us take Then, from (1.3), we have
By (1.1) and (2.3), we see that
Therefore, by (2.4), we obtain the following Lemma.
Lemma 1. For we have
For it is known that
Thus, by (2.5), we get
where are the Bernoulli polynomials of order
In the special case, are called the n-th Bernoulli numbers of order
From (2.4), we note that
Thus, by (2.7), we get
and, from (2.5), we have
Therefore, by (2.8) and (2.9), we obtain the following Lemma.
Lemma 2. For we have
By Lemma 1, we easily see that
(see ), where are the ordinary Bernoulli numbers.
From Lemma 2, we have
(see ), 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
Therefore, by (2.12) and (2.13), we obtain the following Lemma.
Lemma 4. For we have
Remark. For by (2.11), we have
Now, we define the hyper-geometric Daehee polynomials
For example, we have
Thus the hyper-geometric Daehee number are defined by
Note that is the Daehee number.
Therefore, by (2.18), we obtain the following theorem.
Theorem 5. For we have
In (2.17), we have
Therefore, by (2.19), we obtain the following theorem.
Theorem 6. For we have
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