Research Article | Open Access
Novel Identities for -Genocchi Numbers and Polynomials
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 or
We assume that with as an indeterminate. The -adic absolute value , is normally defined by where with , and .
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 -Genocchi polynomials with wegiht are introduced as
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:
Here, we assume that is a fixed parameter. Let Thus, by expression of (2.3), we can readily see the following:
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 (2.3), we easily see
We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.
Theorem 2.1. For , one has
As a special case, we derive the following.
Corollary 2.2. For , one has
Proof. When into (2.10), we derive the following identity: Here, is greatest integer . Then, we complete the proof of Theorem.
From (2.2), we note that
Now, let us consider definition of integral from to in (2.11), then we have where is beta function which is defined by
As a result, we obtain the following theorem.
Theorem 2.3. For , one has
Proof. By taking integral from to in (2.11), we easily reach to desired result.
Substituting into Theorem 2.1, we readily 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
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
Theorem 3.2. For , one has here .
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
Corollary 3.5. For , one gets where
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.
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Copyright © 2012 Serkan Araci. 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.