1. Introduction, Definitions

Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and the classical and the numerical analysis. These polynomials can be defined by various methods depending on the applications. There are six approaches to the theory of Bernoulli polynomials. We prefer here the definition by generating functions given by Euler ^[4].

The classical Bernoulli polynomials and the classical Euler polynomials are defined respectively as

(1.1)

(1.2)

The corresponding Bernoulli numbers and Euler numbers are given by

From (1.1) and (1.2), we easily derive that

(1.3)

(1.4)

(for details, see ^{[11, J. of App. Math., 153-163, 2003.">12, App. Math. Letter, 17, 375-380, 2004.">13]}).

The generalized Apostol-Bernoulli polynomials order and the generalized Apostol-Euler polynomials order are defined respectively by the following generating functions

(1.5)

(1.6)

Recently, Srivastava et. al. in (^{[App. Math. Letter, 17, 375-380, 2004.">13, Russian J. of Math. Phys., 17, 251-261, 2010.">14, Taiwanese J. of Math., 15, 283-305, 2011.">15]}) have investigated some new classes of Apostol-Bernoulli, Apostol-Euler polynomials with parameters a, b, and c. They gave some recurrence relations and proved some theorems.

For one can obtain the classical polynomials (1.1) and (1.2). Other generalizations can be developed as well.

Definition 1. [Natalini ^[12] and S. Chen et al. ^[3]] The generalized Bernoulli polynomials are defined, in a suitable neigbourhood of by means of the generating functions

(1.7)

From (1.7) for we obtain the generating function of classical Bernoulli polynomials From (1.7) for we obtain the generalized Bernoulli numbers

Definition 2. [Kurt ^[9]] For the generalized Bernoulli polynomials of order are defined by means of the generating function

(1.8)

in suitable neigbourhood of

The case was first introduced by Natalini and Bernardini ^[6]. For we obtain classical Bernoulli polynomials.

By the same motivation, the generalized Euler polynomials of order and generalized Euler numbers of order were defined by the author ^[10]

(1.9)

and

(1.10)

From (1.9) and (1.10) and for we obtain classical Euler polynomials and classical Euler numbers respectively:

By the same motivation, the generalized Genocchi polynomials of order and generalized Genocchi numbers of order can be defined as

(1.11)

and

(1.12)

2. New Classes of Generalized Apostol-Euler Polynomials and Apostol-Bernoulli Polynomials

The following definitions provide a natural generalization of the Apostol-Euler polynomials of order and the Apostol-Bernoulli polynomials of order where

Definition 3. We de.ne the generalized Bernoulli polynomials of order and the generalized Euler polynomials of order respectively by

(2.1)

(2.2)

and

(2.3)

For (2.1) reduces to (1.7).

For (2.1), (2.2) and (2.3) reduce to classical Bernoulli polynomial, classical Euler polynomial and classical Genocchi polynomial.

From (2.1), (2.2) and (2.3), we obtain

and

Theorem 1. Let Then the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Euler polynomials satisfy the following relations

(2.4)

(2.5)

and

(2.6)

respectively.

Proof. Considering the generating function (2.1) and comparing the coefficients of in the both sides of the above equation, we arrive at (2.4). Proof of (2.5) and (2.6) are similar.

Theorem 2. The generalized Apostol-Bernoulli polynomials satisfy the following recurrence relation:

(2.7)

Proof. Considering the expression and using generating function (2.1), the proof follows.

Corollary 1. The generalized Apostol-Euler polynomials satisfy the following recurrence relation:

(2.8)

Theorem 3. There is the following relation between the generalized Apostol-Bernoulli polynomials for and the generalized Apostol-Euler polynomials for

(2.9)

Proof. We take and instead of and respectively. We write as:

Comparing the coefficients of , we obtain (2.9).

Theorem 4. The generalized Apostol-Bernoulli polynomials satisfy the following recurrence relation:

(2.10)

Proof. From (2.1) for we write as

(2.11)

(2.12)

We put (2.12) in the right hand side of (2.11). Then

If we make necessary operations in the last equation and comparing the coefficients of we arrive (2.10).

Theorem 5. The following relations hold true:

(2.13)

and

(2.14)

Proof. From (2.1) for

Comparing the coefficients of we obtain (2.13).

For the proof of (2.14), we write

Comparing the coefficients of we arrived to result.

Corollary 2. The new generalized Bernoulli polynomials for and the new generalized Euler polynomials for satisfy the following Raabe relations:

(2.15)

and

(2.16)

Proof. We put in (2.1),

From the last equality, we have (2.15).

Second equation of this corollary can be obtained similarly, so we omit it.

Acknowledgements

This paper was supported by the Scientific Research Project Administration of Akdeniz University.

References

[1]	A. Bagdasaryan and S. Araci, Some new identities on the Apostol-Bernoulli polynomials higher order derived from Bernoulli basis, arXiv:1311.4148 [math.NT].
	In article
[2]	G. Bretti and P. E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. of Math. 8, 415-428, 2004.
	In article
[3]	S. Chen, Yi Chai and Q.-M. Luo, An extension of generalized Apostol-Euler polynomials, Advances in Difference Equation.
	In article
[4]	F. Costabile, F. Dellaccio and M. I. Gualtieri, A new approach to Bernoulli polynomials, Rendi. di. Math. Series VII, 26, 1-12, 2006.
	In article
[5]	S. Gaboury and B. Kurt, Some relations involving Hermite-based Apostol-Genocchi polynomials, App. Math. Sci., 82, 4091-4102, 2012.
	In article
[6]	Y. He and C. Wang, Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials, Discrete Dynamics in Nature and Society, Article ID 927953, 11 pages, 2012.
	In article
[7]	T. Kim, Some identities for the Bernoulli, Euler and Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20, 18-23, 2010.
	In article
[8]	T. Kim, T. Mansour, S.-H. Rim and S.-H. Lee, Apostol-Euler polynomials arising from umbral calculus, Advances in Difference Equations 2013, 2013:300.
	In article
[9]	B. Kurt, A further generalization of the Bernoulli polynomials and on the 2D-Bernoulli polynomials Bn2(x,y), App. Math. Sci., 47, 2315-2322, 2010.
	In article
[10]	B. Kurt, A further generalization of the Euler polynomials and on the 2D-Euler polynomials, Proc. Jang. Math. Soc., 15, 389-394, 2012.
	In article
[11]	Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Int. Trans. Spec. Func. Vol 20(5), 377-391, 2009.
	In article		CrossRef
[12]	P. Natalini and A. Bernardini, A Generalization of the Bernoulli polynomials, J. of App. Math., 153-163, 2003.
	In article
[13]	H. M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, App. Math. Letter, 17, 375-380, 2004.
	In article		CrossRef
[14]	H. M. Srivastava, M. Garg and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. of Math. Phys., 17, 251-261, 2010.
	In article		CrossRef
[15]	H. M. Srivastava, M. Garg and S. Choudhary, Some new families of neralized Euler and Genocchi polynomials, Taiwanese J. of Math., 15, 283-305, 2011.
	In article
[16]	R. Trembly, S. Gaboury and B.-J. Fugére, A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pinter addition theorem, Applied Math. Letter, 24, 1888-1893, 2011.
	In article		CrossRef
[17]	R. Trembly, S. Gaboury and B.-J. Fugére, Some new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials, Inter. J. of Math. and Math. Sci., 2012.
	In article