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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 298531, 11 pages
Explicit Formulas Involving -Euler Numbers and Polynomials
1Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey
2Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
Received 4 April 2012; Accepted 4 October 2012
Academic Editor: Gerd Teschke
Copyright © 2012 Serkan Araci et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.
Imagine that is a fixed odd prime number. Throughout this paper we use the following notations, where denotes the ring of -adic rational integers, denotes the field of rational numbers, denotes the field of -adic rational numbers, and denotes the completion of algebraic closure of . Let be the set of natural numbers and .
The -adic absolute value is defined by
In this paper, we will assume that as an indeterminate.
We say that is a uniformly differentiable function at a point , if the difference quotient has a limit as and denote this by .
Let be the set of uniformly differentiable function on . For , let us start with the expression which represents -adic -analogue of Riemann sums for . The integral of on will be defined as the limit of these sums, when it exists. The -adic -integral of function is defined by Kim
In , the -Euler polynomials with weight are introduced as
From (1.7), we have where are called -Euler numbers with weight . Then, -Euler numbers are defined as where the usual convention about replacing by is used.
Similarly, the -Bernoulli polynomials and numbers with weight are defined, respectively, as (for more information, see ).
We, by using the Kim et al. method in , will investigate some interesting identities on the -Euler numbers and polynomials arising from their generating function and derivative operator. Consequently, we derive some properties on the -Euler numbers and polynomials and -Bernoulli numbers and polynomials by using -Volkenborn integral and fermionic -adic -integral on .
2. On the -Euler Numbers and Polynomials
Let us consider Kim’s -Euler polynomials as follows:
Here is a fixed parameter. Thus, by expression of (2.1), we can readily see the following:
Let us take derivative operator on both sides of (2.4). Then we get
Let (not ) be the constant term in a Laurent series of . Then, from (2.5), we get
By (2.1), we see
From (2.1), we note that
By (2.9), we easily see
Now, let us consider definition of integral from to in (2.8), then we have where is beta function which is defined by
As a result, we obtain the following theorem.
Theorem 2.1. For , one has
Substituting into Theorem 2.1, we readily get
By (2.1), it follows that
Let in (2.1), we see that
Last from equality, we discover the following:
Here is Gauss’ symbol. Then, taking integral from to in both sides of last equality, we get
Consequently, we derive the following theorem.
Theorem 2.2. The following identity is true.
Theorem 2.3. For , one has
3. -adic Integral on Associated with Kim’s -Euler Polynomials
In this section, we consider Kim’s -Euler polynomials by means of -adic -integral on . Now we start with the following assertion.
Let . Then by (2.8),
On the other hand, in right hand side of (2.8),
Equating and , we get the following theorem.
Theorem 3.1. For , one has
Let us take fermionic -adic -integral on in left hand side of (2.21), we get
In other words, we consider right hand side of (2.21) as follows:
Equating and , we get the following theorem.
Theorem 3.2. For , one has
On the other hand,
Therefore, we get the following theorem.
Theorem 3.3. For , one has
By using fermionic -adic -integral on in left hand side of (2.21), we get
Also, we consider right hand side of (2.21) as follows:
Equating and , we get the following corollary.
Corollary 3.4. For , one gets
The authors would like to thank the referee for his/her valuable comments on this work.
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