Identities on the Bernoulli and Genocchi Numbers and Polynomials
Seog-Hoon Rim,1Joohee Jeong,1and Sun-Jung Lee2
Academic Editor: Yilmaz Simsek
Received09 Jun 2012
Accepted09 Aug 2012
Published11 Sept 2012
Abstract
We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.
1. Introduction
Let be a fixed odd prime number. Throughout this paper , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of . Let be the set of natural numbers and . The -adic norm on is normalized so that . Let be the space of continuous functions on . For , the fermionic -adic integral on is defined by Kim as follows:
(see [1–16]). From (1.1), we have
(see [1–16]), where .
Let us take . Then, by (1.2), we get
where are the th ordinary Genocchi numbers (see [8, 15]).
From the same method of (1.3), we can also derive the following equation:
where are called the th Genocchi polynomials (see [14, 15]).
By (1.3), we easily see that
(see [15]). By (1.3) and (1.4), we get Witt's formula for the th Genocchi numbers and polynomials as follows:
From (1.2), we have
where the symbol is the Kronecker symbol (see [4, 5]).
Thus, by (1.5) and (1.7), we get
(see [15]). From (1.4), we can derive the following equation:
By (1.6) and (1.9), we see that
Thus, by (1.10), we get .
From (1.5) and (1.8), we have
The Bernoulli polynomials are defined by
(see [6, 9, 12]) with the usual convention about replacing by .
In the special case, , is called the -th Bernoulli number. By (1.12), we easily see that
(see [6]). Thus, by (1.12) and (1.13), we get reflection symmetric formula for the Bernoulli polynomials as follows:
(see [6, 9, 12]). From (1.14) and (1.15), we can also derive the following identity:
In this paper, we investigate some properties of the fermionic -adic integrals on . By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics.
2. Identities on the Bernoulli and Genocchi Numbers and Polynomials
Let us consider the following fermionic -adic integral on as follows:
On the other hand, by (1.14) and (1.15), we get
Equating (2.1) and (2.2), we obtain the following theorem.
Theorem 2.1. For , one has
By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers.
Now, we consider the fermionic -adic integral on for the polynomials as follows:
On the other hand, by (1.8), (1.10), and (1.11), we get
Equating (2.4) and (2.5), we obtain the following theorem.
Theorem 2.2. For , one has
Let us consider the fermionic -adic integral on for the product of and as follows:
On the other hand, by (1.10) and (1.14), we get
By (2.7) and (2.8), we easily see that
Therefore, by (2.9), we obtain the following theorem.
Theorem 2.3. For , one has
Corollary 2.4. For , one has
Let us consider the fermionic -adic integral on for the product of the Bernoulli polynomials and the Bernstein polynomials. For , with , are called the Bernstein polynomials of degree , see [11]. It is easy to show that ,
On the other hand, by (1.14) and (2.12), we get
Equating (2.12) and (2.13), we see that
Thus, from (2.14), we obtain the following theorem.
Theorem 2.5. For , one has
Finally, we consider the fermionic -adic integral on for the product of the Euler polynomials and the Bernstein polynomials as follows:
On the other hand, by (1.10) and (2.12), we get
Equating (2.16) and (2.17), we obtain
Therefore, by (2.18), we obtain the following theorem.
Theorem 2.6. For , one has
Acknowledgment
This paper was supported by Kynugpook National University Research Fund, 2012.
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