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Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums
Advances in Difference Equations volume 2013, Article number: 32 (2013)
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.
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]:
The generalized Apostol-Bernoulli polynomials of order and the generalized Apostol-Euler polynomials of order are defined respectively by the following generating functions [3]:
Recently, Garg et al. in [4] introduced the following generalization of the Hurwitz-Lerch zeta function :
(, a, , , when ; and when ). It is obvious that
(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:
From (6) and (7), we have
and
(see [6]).
From (8) and (9), for , we have respectively
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 2.1 There is the following relation between Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function :
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 2.2 For all , , we have the following symmetry identity:
Proof Let . Then
From (3) and (8), we have
In a similar manner,
Comparing the coefficients of , we proved the theorem. □
Corollary 2.3 We put in (13). We have
3 Some symmetry identities for the Apostol-Euler polynomials
Theorem 3.1 Let a and b be positive integers with the same parity. Then
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 3.2 Let a and b be positive integers with the same parity. Then
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 3.3 Let p, l, a, b and n be positive integers and a, b be of the same parity. Then
Proof Let . From (3) and (10), we have
On the other hand, we write the function as
Equating the coefficient of , we obtain (16). □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
This paper was supported by the Scientific Research Project Administration of Akdeniz University.
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Kurt, V. Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums. Adv Differ Equ 2013, 32 (2013). https://doi.org/10.1186/1687-1847-2013-32
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DOI: https://doi.org/10.1186/1687-1847-2013-32