The main purpose of this paper is to study on generating functions of the Genocchi numbers and polynomials. We prove new relation for the generalized Genocchi numbers which is related to the Genocchi numbers and Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define Genocchi zeta and functions, which are interpolated Genocchi numbers and polynomials at negative integers. We also give some applications of generalized Genocchi numbers.
1. Introduction Definitions and Notations
In [1], Jang et al. gave new formulae on Genocchi numbers. They defined polyGenocchi numbers to give the relation between Genocchi numbers, Euler numbers, and polyGenocchi numbers. In [2], Kim et al. constructed new generating functions of the analogue Eulerian numbers and analogue Genocchi numbers. They gave relations between Bernoulli numbers, Euler numbers, and Genocchi numbers. They also defined Genocchi zeta functions which interpolate these numbers at negative integers. Kim [3] gave new concept of the extension of Genocchi numbers and gave some relations between Genocchi polynomials and Euler numbers. In this paper, by using generating function of this numbers, we study on Genocchi zeta and functions. In [4], Kim constructed Genocchi numbers and polynomials. By using these numbers and polynomials, he proved the analogue of alternating sums of powers of consecutive integers due to Euler:
(cf. [4]), where if
and the numbers are called Genocchi numbers which are defined by
Note that (cf. [3, 5–9]). The Euler numbers are usually defined by means of the following generating function (cf. [10–16]):
The Genocchi numbers are usually defined by means of the following generating function (cf. [12, 13]):
These numbers are classical and important in number theory. In [12], Kim defined generating functions of the Genocchi numbers and Euler numbers as follows:
where denotes Euler numbers,
where denotes Genocchi numbers. Genocchi zeta function is defined as follows (cf. [13, page 108]): for
Kim [17] defined the ferminoic and deformic expression of adic Volkenborn integral at and He constructed integral equation of the fermionic expression of adic Volkenborn integral at By using this integral equation, he defined new generating functions of Euler numbers and polynomials. By using derivative operator to this functions, he constructed new zeta, functions and adic functions, which are interpolated Euler numbers and polynomials. He also gave some applications which are the formulae of the trigonometric functions by applying ferminoic and deformic expression of adic Volkenborn integral at and Kim and Rim [18] defined twovariable function. They gave main properties of this function. In [6], Kim constructed the twovariable adic function which interpolates the generalized Bernoulli polynomials attached to Dirichlet character. In [19], Simsek et al. constructed the twovariable Dirichlet function and the twovariable multiple Dirichlettype Changhee function. In [8, 20], Simsek defined generating functions, which are interpolates twisted Bernoulli numbers and polynomials, twisted Euler numbers and polynomials. He[21] also gave new generating functions which produce Genocchi zeta functions and series with attached to Dirichlet character. Therefore, by using these generating functions, he constructed new analogue of HardyBerndt sums. He gave relations between these sums, Genocchi zeta functions and series as well,
(cf. [21]), where is Euler's gamma function and (cf. [1], [13, page 108, equation (2.43)]). The first author defined analogue of the Genocchi zeta functions as follows [21].
Definition 1.1.
Let and . analogue of the Genocchi type zeta function is expressed by the formula
Remark 1.2.
If then (1.10) reduces to ordinary Genocchi zeta functions (see [13, page 108]). Cenkci et al. [22], defined different type of Genocchi zeta functions, which are defined as follows:
Simsek [21] defined analogue of the Hurwitztype Genocchi zeta function by applying the Mellin transformations as follows:
Definition 1.3 ({see [21]}).
Let , and . analogue of the Hurwitztype Genocchi zeta function is expressed by the formula
Observe that when the is reduced to and if then A function is called an ordinary Hurwitztype Genocchi zeta function if is expressed by the formula
where , , and , cf. [13].
In [21], Simsek defined analogue (Genocchitype) one and twovariable functions as follows, respectively; let be a Dirichlet character; let and
A function is called an ordinary Genocchitype function if is expressed by the formula
where and cf. [13].
Observe that when (1.15) reduces to (1.10):
We summarize our work as follows. In Section 2, we study on generating functions of the Genocchi numbers and polynomials. By using infinite and finite series, we give some definitions of the Genocchi numbers and polynomials. We find new relations between generalized Genocchi numbers with attached to Genocchi numbers and Barnes' type Changhee Bernoulli numbers. In Section 3, by applying Mellin transformation and derivative operator to the generating functions of the Genocchi numbers, we construct Genocchi zeta and functions, which are interpolated Genocchi numbers and polynomials at negative integers. We also give some new relations related to these numbers and polynomials.
2. genocchi Number and Polynomials
In this section, we give some new relations and identities related to Genocchi numbers and polynomials. Firstly we give some generating functions of the Genocchi numbers, which were defined by Kim [3, 10, 11]:
and let
(cf.[3, 10, 11, 23]), where denotes Genocchi numbers.
We note that Genocchi numbers, were defined by Kim [3, 10, 11].
By using the above generating functions, Genocchi polynomials, , are defined by means of the following generating function:
Our generating function of is similar to that of [3, 12, 21, 23]. By using Cauchy product in (2.3), we easily obtain
Then by comparing coefficients of on both sides of the above equation, for we obtain the following result.
Theorem 2.1.
Let be an integer with Then one has
By using the same method in [3, 12, 21] in (2.3), we have
and after some elementary calculations, we have
By comparing coefficients of on both sides of the above equation, we arrive at the following corollary.
Corollary 2.2.
Let . Then one has
We give some of Genocchi polynomials as follows:
From the generating function we have the following.
Corollary 2.3.
Let . Then one has
Proof of the Corollary 2.3 was given by Kim [3, 12]. We give some of Genocchi numbers as follows: ,
Observe that if then
By using derivative operator to (2.6), we have
After some elementary calculations, we arrive at the following corollary.
Corollary 2.4.
Let be a positive integer. Then one has
Corollary 2.5.
Let be a positive integer. Then one has
Proof.
Proof of this corollary is easily obtained from (2.4).
Generalized Genocchi numbers are defined by means of the following generating function (this generating function is similar to that of [3, 12, 21–24]):
where denotes the Dirichlet character with conductor the set of positive integers.
Observe that when (2.13) reduces to (2.3).
By (2.13), we have
After some elementary calculations and by comparing coefficients on both sides of the above equation, we get
By setting , where and , in the above equation, we obtain
In [15], Srivastava et al. defined the following generalized Barnestype Changhee Bernoulli numbers.
Let be the Dirichlet character with conductor . Then the generalized Barnestype Changhee Bernoulli numbers with attached to are defined as follows:
(cf. [15]). Substituting and into the above equation, we have
By using derivative operator to the above, we obtain
By substituting (2.9) and (2.19) into (2.16), after some calculations, we arrive at the following theorem.
Theorem 2.6.
Let be the Dirichlet character with conductor . If is odd, then one has
if is even, then one has
where is defined in (2.19).
Remark 2.7.
In Theorem 2.6, we give new relations between generalized Genocchi numbers, with attached to , Genocchi numbers, , and Barnestype Changhee Bernoulli numbers. For detailed information about generalized Barnestype Changhee Bernoulli numbers with attached to see [15].
Generalized Genocchi polynomials are defined by means of the following generating function:
Theorem 2.8.
Let be the Dirichlet character with conductor . Then one has
Remark 2.9.
Generating functions of and are different from those of [3, 12, 22, 23]. Kim defined generating function of as follows [12]:
In [21], Simsek defined generating function of by
3. genocchi Zeta and Functions
In recent years, many mathematicians and physicians have investigated zeta functions, multiple zeta functions, series, Genocchi zeta, and functions, and Bernoulli, Euler, and Genocchi numbers and polynomials mainly because of their interest and importance. These functions and numbers are not only used in complex analysis, but also used in adic analysis and other areas. In particular, multiple zeta functions occur within the context of Knot theory, quantum field theory, applied analysis and number theory, (cf. [15]). In this section, we define Genocchi zeta and functions, which are interpolated Genocchi polynomials and generalized Genocchi numbers at negative integers. By applying the Mellin transformation to (2.3), we obtain
where , and .
Thus, Hurwitztype Genocchi zeta function is defined by the following definition.
Definition 3.1.
Let with and let with Thenone defines
Observe that when in (3.2), then we obtain Riemanntype Genocchi zeta function:
Hurwitztype Genocchi zeta function interpolates Genocchi polynomials at negative integers. For , , and by applying Cauchy residue theorem to (3.1), we can obtain the following theorem.
Theorem 3.2.
For , then one has
Remark 3.3.
The second proof of Theorem 3.2 can be obtained by using derivative operator to (2.3) as follows:
Thus we obtained the desired result.
By applying Mellin transformation to (2.13), we obtain
Thus we can define Dirichlettype Genocchi function as follows.
Definition 3.4.
Let be the Dirichlet character with conductor . Let with One defines
Relation between and is given by the following theorem.
Theorem 3.5.
Let be the Dirichlet character with conductor . Then one has
Proof.
By setting where in (3.7), we obtain,
After some elementary calculations, we arrive at the desired result of the theorem.
The function interpolates generalized Genocchi numbers, which are given by the following theorem.
Theorem 3.6.
Let Let be the Dirichlet character with conductor Then one has
Proof.
Proof of this theorem is similar to that of Theorem 3.2. So we omit the proof.
We give some applications. Setting and using Theorem 3.2 in Theorem 3.5, we get
By comparing both sides of the above equation and Theorem 3.6, we obtain distributions relation of the generalized Genocchi numbers as follows.
Corollary 3.7.
Let be the Dirichlet character with conductor Then one has
where and is the Genocchi polynomial.
By substituting (2.5) into (3.12), we have the following corollary.
Corollary 3.8.
Let be the Dirichlet character with conductor . Then one has
If we substitute (2.7) into (3.12), we get a new relation for the distribution relation of Genocchi numbers:
Thus we arrive at the following corollary.
Corollary 3.9.
Let be the Dirichlet character with conductor . Then one has
Acknowledgments
The first and third authors have been supported by the Scientific Research Project, Administration Akdeniz University. The second author has been supported by Uludag University Research Fund, Projects no. F2004/40 and F200831. The fourth author has been supported by National Institute for Mathematical Sciences Doryongdong, Yuseonggu, Daejeon. The authors express their sincere gratitude to referees for their suggestions and comments.
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