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# Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums

Veli Kurt

Author Affiliations

Department of Mathematics, Faculty of Science, Akdeniz University, Campus, Antalya, 07058, Turkey

Advances in Difference Equations 2013, 2013:32  doi:10.1186/1687-1847-2013-32

 Received: 9 November 2012 Accepted: 22 January 2013 Published: 11 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In recent years, symmetry properties of the Bernoulli polynomials and the Euler polynomials have been studied by a large group of mathematicians (He and Wang in Discrete Dyn. Nat. Soc. 2012:927953, 2012, Kim et al. in J. Differ. Equ. Appl. 14:1267-1277, 2008; Abstr. Appl. Anal. 2008, doi:11.1155/2008/914347, Yang et al. in Discrete Math. 308:550-554, 2008; J. Math. Res. Expo. 30:457-464, 2010). Luo (Integral Transforms Spec. Funct. 20:377-391, 2009), introduced the lambda-multiple power sum and proved the multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order. Ozarslan (Comput. Math. Appl. 2011:2452-2462, 2011), Lu and Srivastava (Comput. Math. Appl. 2011, doi:10.1016/j.2011.09.010.2011) gave some symmetry identities relations for the Apostol-Bernoulli and Apostol-Euler polynomials.

In this work, we prove some symmetry identities for the Apostol-Bernoulli and Apostol-Euler polynomials related to multiple alternating sums.

AMS Subject Classification: 11F20, 11B68, 11M35, 11M41.

##### Keywords:
Bernoulli polynomials; Euler polynomials; Apostol-Bernoulli polynomials; Apostol-Euler polynomials; symmetry relation; power sums; alternating sums

### 1 Introduction, definitions and notations

The generalized Bernoulli polynomials of order and the generalized Euler polynomials of order , each of degree n as well as in α, are defined respectively by the following generating functions [1-3]:

(1)

(2)

The generalized Apostol-Bernoulli polynomials of order and the generalized Apostol-Euler polynomials of order are defined respectively by the following generating functions [3]:

(3)

(4)

Recently, Garg et al. in [4] introduced the following generalization of the Hurwitz-Lerch zeta function :

(, a, , , when ; and when ). It is obvious that

(5)

(for details on this subject, see [3-5]).

The multiple power sums and the λ-multiple alternating sums are defined by Luo [6] as follows:

(6)

(7)

From (6) and (7), we have

(8)

and

(9)

(see [6]).

From (8) and (9), for , we have respectively

(10)

(11)

Symmetry property and some recurrence relations of the Bernoulli polynomials, Euler polynomials, Apostol-Bernoulli polynomials and Apostol-Euler polynomials have been investigated by a lot of mathematicians [1-24]. Firstly, Yang [22] proved symmetry relation for Bernoulli polynomials. Wang et al. in [1,20,21] gave some symmetry relations for the Apostol-Bernoulli polynomials. Kim in [8,10,11,14,15] proved symmetric identities for the Bernoulli polynomials and Euler polynomials. Luo in [6,17] gave multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials. Also, he defined λ-power sums. Srivastava et al.[2,3,5] proved some theorems and relations for these polynomials. They proved some symmetry identities for these polynomials.

In this work, we give some symmetry identities for the Apostol-type polynomials related to multiple alternating sums.

### 2 Symmetry identities for the Apostol-Bernoulli polynomials

We will prove the following theorem for the Apostol-Euler polynomials, which are symmetric in a and b.

Theorem 2.1There is the following relation between Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function:

(12)

Proof Let . Then

From (3) and (10), we write

where . After the Cauchy product, we have

In a similar manner,

From (3) and (10), we write

Since , after the Cauchy product, we have

Compressing to coefficients and by using (5), we prove the theorem. □

Theorem 2.2For all, , we have the following symmetry identity:

(13)

Proof Let . Then

From (3) and (8), we have

In a similar manner,

Comparing the coefficients of , we proved the theorem. □

Corollary 2.3We putin (13). We have

### 3 Some symmetry identities for the Apostol-Euler polynomials

Theorem 3.1Letaandbbe positive integers with the same parity. Then

(14)

Proof Let . From (4) and (9) for , we have

Since , the expression for is symmetric in a and b. Therefore, we obtain the following power series for by symmetry:

Equating the coefficient of in the two expressions for gives us the desired result. □

Theorem 3.2Letaandbbe positive integers with the same parity. Then

(15)

Proof Let . From (4) and (9), we write

Since , the expression for is symmetric in a and b.

In a similar manner, we have

Equating the coefficient of in the two expressions for gives us the desired result. □

Theorem 3.3Letp, l, a, bandnbe positive integers anda, bbe of the same parity. Then

(16)

Proof Let . From (3) and (10), we have

On the other hand, we write the function as

Equating the coefficient of , we obtain (16). □

### Competing interests

The author declares that they have no competing interests.

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

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