Dae San Kim,1Dmitry V. Dolgy,2Hyun-Mee Kim,3Sang-Hun Lee,3and Taekyun Kim4
Academic Editor: Lee Chae Jang
Received24 Feb 2012
Accepted10 May 2012
Published27 Jun 2012
Abstract
Recently, some interesting and new identities are introduced in (Hwang et al., Communicated). From these identities, we derive some new and interesting integral formulae for the Bernoulli polynomials.
1. Introduction
As is well known, the Bernoulli polynomials are defined by generating functions as follows:
(see [1–11]). In the special case, are called the th Bernoulli numbers. The Euler polynomials are also defined by
with the usual convention about replacing by (see [1–11]). From (1.1) and (1.2), we can easily derive the following equation:
By (1.1) and (1.3), we get
From (1.1), we have
Thus, by (1.5), we get
It is known that are called the th Euler numbers (see [7]). The Euler polynomials are also given by
(see [6]). From (1.7), we can derive the following equation:
By the definition of Bernoulli and Euler numbers, we get the following recurrence formulae:
where is the kronecker symbol (see [5]). From (1.6), (1.8), and (1.9), we note that
where . The following identity is known in [5]:
From the identities of Bernoulli polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli polynomials.
Therefore, by (2.2) and (2.15), we obtain the following theorem.
Theorem 2.5. For , one has
3. -Adic Integral on Associated with Bernoulli and Euler Numbers
Let be a fixed odd prime number. Throughout this section, , , and will denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the normalized exponential valuation of with . Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by
(see [8]). Thus, by (3.1), we get
where , and . Let us take . Then we have
From (3.3), we have
From (1.2), we can derive the following integral equation:
Thus, from (3.4) and (3.5), we get
From (3.6), we have
The fermionic -adic integral on is defined by Kim as follows [6, 7]:
Let . Then we have
Continuing this process, we obtain the following equation:
Thus, by (3.10), we have
Let us take . By (3.9), we get
From (3.2), we have the Witt's formula for the th Euler polynomials and numbers as follows:
By (3.11) and (3.13), we get
Let us consider the following -adic integral on :
From (1.4) and (3.15), we have
Therefore, by (3.15) and (3.16), we obtain the following theorem.
Theorem 3.1. For , one has
Now, we set
By (1.4), we get
Therefore, by (3.18) and (3.19), we obtain the following theorem.
Theorem 3.2. For , one has
Let us consider the following integral on :
From (2.2), we have
Therefore, by (3.21) and (3.22), we obtain the following theorem.
Theorem 3.3. For , one has
Now, we set
By (2.2), we get
Therefore, by (3.24) and (3.25), we obtain the following corollary.
Corollary 3.4. For , we have
Let us assume that . From Lemma 2.4 and (3.13), we note that
Thus, by (3.28) and (3.13), we obtain the following lemma (see [5]).
Lemma 3.5. Let . For , one has
Let us consider the formula in Lemma 3.5 with . Then we have
Taking on both sides of (3.30),
By the same method, we get
Therefore, by (3.31) and (3.32), we obtain the following proposition.
Proposition 3.6. Let . Then one has
Replacing by , we have
From (3.4) and (3.7), we derive some identity for the first term of the LHS of (3.34).
The first term of the LHS of (3.34)
where
The second term of the LHS of (3.34)
Therefore, by (3.34), (3.35), and (3.37), we obtain the following theorem.
Theorem 3.7. Let with . Then one has
where
Remark 3.8. Here, we note that
Acknowledgment
The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.
References
A. Bayad and T. Kim, “Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 2, pp. 247–253, 2010.
D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.
I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order --Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
K.-W. Hwang, D. V. Dolgy, T. Kim, and S. H. Lee, “On the higher-order -Euler numbers and polynomials with weight ,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 354329, 12 pages, 2011.
T. Kim, “Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.
T. Kim, “Symmetry -adic invariant integral on for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.
H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on -Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
C. S. Ryoo, “Some identities of the twisted -Euler numbers and polynomials associated with -Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011.
Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.