The essential aim of this paper is to introduce novel identities for q-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related to p-adic analogue of Bernstein polynomials. Also, we derive relations between q-Genocchi and q-Bernoulli numbers.
Imagine that be a fixed odd prime number. We now start with definition of the following notations. Let be the field -adic rational numbers and let be the completion of algebraic closure of .
Then is integral domain, which is defined by
We assume that with as an indeterminate. The -adic absolute value , is normally defined by
where with , and .
is a -extension of , which is defined by
we note that (see [1–17]).
Throughout this paper, we use notation of , where denotes set of Natural numbers.
We say that is a uniformly differentiable function at a point , if the difference quotient,
has a limit as , and we denote this by . Then, for , we can start with the following expression:
which represents a -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 [7, 12] as
The bosonic integral is considered as a bosonic limit , . Similarly, the fermionic -adic integral on is introduced by Kim as follows:
(for more details, see [13–16]).
From (1.9), it is well-known equality that
where (for details, see [2, 3, 8, 9, 12, 13, 15–17]).
The -Genocchi polynomials with wegiht are introduced as
From (1.11), we have
where are called -Genocchi numbers with weight . Then, -Genocchi numbers are defined as
with the usual convention about replacing by is used (for details, see ).
Let be the space of continuous functions on . For , -adic analogue of Bernstein operator for is defined by
where . Here, is called -adic Bernstein polynomials, which are defined by
(for details, see [1, 4, 5, 7]).
The -Bernoulli polynomials and numbers with weight are defined by Kim et al., respectively,
(for more information, see ).
The author, by using derivative operator, will investigate some interesting identities on the -Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between -Genocchi numbers and -Bernoulli numbers by using Kim’s -Volkenborn integral and fermionic -adic -integral on .
2. Novel Properties of -Genocchi Numbers and Polynomials with Weight 0
Let . Then, by using (1.10), we easily procure the following:
From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s -Genocchi polynomials with weight in  as follows:
Here, we assume that is a fixed parameter. Let
Thus, by expression of (2.3), we can readily see the following:
Last from equality, taking derivative operator as on the both sides of (2.4), then, we easily see that
where and is identity operator. By multiplying on both sides of (2.5), we get
Let us take derivative operator on the both sides of (2.6). Then, we get
Let (not ) be the constant term in a Laurent series of in (2.3). Then, we get
By differentiating both sides of (2.11) with respect to , we have the following:
We now give interesting theorem for -Genocchi numbers with weight as follows.
Theorem 2.4. For , one has
Proof. It is proved by using definition of integral on the both sides in the following equality, that is,
Last from equality, we discover the following:
Then, taking integral from to both sides of last equality, we get
Thus, we complete the proof of the theorem.
Theorem 2.5. For , one has
Proof. In view of (2.2) and (2.23), we discover the following applications:
Thus, we give evidence of the theorem.
As into Theorem 2.5, it leads to the following interesting property.
Corollary 2.6. For , one has
where is ordinary Genocchi polynomials, which is defined by the means of the following generating function :
3. Some Identities -Genocchi Numbers and -Bernoulli Numbers by Using Kim’s -Adic -Integrals on
In this section, we consider -Genocchi numbers and -Bernoulli numbers by means of -adic -integral on . Now, we start with the following theorem.
Theorem 3.1. For ,, one has
Proof. For , then by (2.11),
On the other hand, the right hand side of (2.11),
Combining and , we arrive to the proof of the theorem.
Theorem 3.2. For , one has
Proof. Let us take fermionic -adic -inetgral on left-hand side of Theorem 2.5, we get
In other word, we consider the right-hand side of Theorem 2.5 as follows:
Equating and , we complete the proof of the theorem.
As in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.
Corollary 3.3. For , one has
Theorem 3.4. For , one has
Proof. We consider (2.11) and (2.2) by means of -Volkenborn integral. Then, by (2.11), we see
On the other hand,
Therefore, we get the proof of theorem.
Corollary 3.5. For , one gets
Proof. By using -adic -integral on left-hand side of Theorem 2.5, we get
Also, we compute the right-hand side of Theorem 2.5 as follows:
Equating and , we get the proof of Corollary.
As in the above theorem, we easily derive the following corollary.
Corollary 3.6. For , one has
The author would like to express his sincere gratitude to the referee for his/her valuable comments and suggestions which have improved the presentation of the paper.
S. Araci, D. Erdal, and J.-J. Seo, “A study on the fermionic -adic -integral representation on associated with weighted -bernstein and -genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011.
S. Araci, J. J. Seo, and D. Erdal, “New construction weighted -Genocchi numbers and polynomials related to zeta type functions,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 487490, 7 pages, 2011.View at Publisher · View at Google Scholar
S. Araci, M. Acikgoz, and F. Qi, “On the q-Genocchi numbers and polynomials with weight zero and their applications,” Number Theory, http://arxiv.org/abs/1202.2643.
T. Kim, B. Lee, S. H. Lee, and S.-H. Rim, “Identities for the Bernoulli and Euler numbers and polynomials,” Ars Combinatoria. In press.
D. S. Kim, T. Kim, S.-H. Lee, D. V. Dolgy, and S.-H. Rim, “Some new identities on the Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 856132, 11 pages, 2011.View at Publisher · View at Google Scholar
T. Kim, J. Choi, and Y. H. Kim, “Some identities on the -Bernoulli numbers and polynomials with weight 0,” Abstract and Applied Analysis, vol. 2011, Article ID 361484, 8 pages, 2011.View at Publisher · View at Google Scholar
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.View at Publisher · View at Google Scholar
D. Kim, T. Kim, J. Choi, and Y.-H. Kim, “Identities involving q-Bernoulli and q-Euler numbers,” Abstract and Applied Analysis, vol. 2012, Article ID 674210, 10 pages, 2012.View at Publisher · View at Google Scholar
C. S. Ryoo, “A note on the weighted q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, pp. 47–54, 2011.