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 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

(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 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

(1.7)

(see ^[13]),

where the factor of in the denominator is present for historical reasons of notation, and the resulting generalized hyper-geometric function is written

(1.8)

(see ^[13]).

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:

(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 n-th 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 hyper-geometric Daehee polynomials

(2.14)

where

For example, we have

(2.15)

Thus the hyper-geometric 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

References

[1]	S. Araci, M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 399-406.
	In article
[2]	A. Bayad, Special values of Lerch zeta function and their Fourier expansions, Adv. Stud. Contemp. Math. 21 (2011), no. 1, 1-4.
	In article
[3]	L. Carlitz, A note on Bernoulli and Euler polynomials of the second kind, Scripta Math. 25 (1961), 323-330.
	In article
[4]	L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.
	In article		CrossRef
[5]	H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79(1972), 44-51.
	In article		CrossRef
[6]	D. S. Kim,T. Kim, Daehee Numbers and Polynomials, Applied Mathematical Sciences, Vol. 7, 2013, no. 120, 5969-5976
	In article
[7]	T. Kim, D. S. Kim, T. Mansour, S. H. Rim, M. Schork, Umbral calculus and Sheffer sequence of polynomials, J. Math. Phys. 54, 083504 (2013) : https://dx.doi.org/10.1063/1.4817853 (15 pages).
	In article
[8]	T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms Spec. Funct. 13 (2002), no. 1, 65-69.
	In article
[9]	V. Kurt, Some relation between the Bernstein polynomials and second kind Bernoulli polynomials, Adv. Stud. Contemp. Math. 23 (2013), no. 1, 43-48.
	In article
[10]	H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18 (2009), no. 1, 41-48.
	In article
[11]	Y. Simsek, S-H. Rim, L. -C. Jang, D.-J. Kang, J.-J. Seo, A note q-Daehee sums, Proceedings of the 16th International Conference of the Jangjeon Mathematical Society, 159-166, Jangjeon Math. Soc., Hapcheon, 2005.
	In article
[12]	Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (2010), no. 4, 495-508.
	In article
[13]	Koepf. W, Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities, Braunschweig, Germany, Vieweg, 1998.
	In article