Keywords: Bernoulli polynomials and numbers, ApostolBernoulli polynomials, ApostolEuler polynomials, Generalized ApostolBernoulli polynomials
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
pp 5458.
DOI: 10.12691/tjant1111
Received November 28, 2013; Revised November 30, 2013; Accepted December 03, 2013
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
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, 12, 13]}).
The generalized ApostolBernoulli polynomials order and the generalized ApostolEuler polynomials order are defined respectively by the following generating functions
 (1.5) 
 (1.6) 
Recently, Srivastava et. al. in (^{[13, 14, 15]}) have investigated some new classes of ApostolBernoulli, ApostolEuler 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 ApostolEuler Polynomials and ApostolBernoulli Polynomials
The following definitions provide a natural generalization of the ApostolEuler polynomials of order and the ApostolBernoulli 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 ApostolBernoulli polynomials and the generalized ApostolEuler 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 ApostolBernoulli 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 ApostolEuler polynomials satisfy the following recurrence relation:
 (2.8) 
Theorem 3. There is the following relation between the generalized ApostolBernoulli polynomials for and the generalized ApostolEuler 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 ApostolBernoulli 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.
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