Reduction and duality of the generalized Hurwitz-Lerch zetas
© Bayad and Chikhi; licensee Springer 2013
Received: 14 December 2012
Accepted: 15 March 2013
Published: 4 April 2013
In this paper, by means of integral representation, we introduce the generalized Hurwitz-Lerch zeta functions of arbitrary complex order. For these functions, we establish the reduction formula and its associated dual formula. We then investigate analytic continuations to the whole complex plane and special values. By means of these reduction and dual formulas, we obtain nice and useful formulas for the Bernoulli-Nörlund and Apostol-Euler-Nörlund polynomials.
The origin of the Hurwitz-Lerch zeta functions and their study go back to Riemann and Hurwitz. In fact, these zeta functions have many important identities which are at the origin of numerous applications in various areas in mathematics and physics. In this paper we introduce and investigate reduction and duality formulas for the generalized Hurwitz-Lerch zeta functions . As an application, we show how these formulas can be easily used for the study of the convolution relations and computation of special values of the Apostol-Bernoulli and Apostol-Euler-Nörlund polynomials of an arbitrary order.
Throughout this paper, we use the following notations, definitions and identities.
1.1 Notations and preliminaries
where is the Kronecker delta symbol.
The ordinary Hurwitz-Lerch zeta function , which corresponds to the function , was originally defined in  by Erdelyi et al. Moreover, Choi and Srivastava [5–7], Kanemitsu et al.  and Nakamura  presented its various properties and applications.
for any non-negative integer n. Therefore, by means of equations (8), (10) and (11), we can easily write as a linear combination of the Bernoulli polynomials , with .
In this paper we deal with the following. Replacing the integer N by any complex number α, we relax the definition of the multiple Hurwitz-Lerch zeta function, and we generalize the formulas (8), (10) and (11). We prove reduction and duality formulas for the Hurwitz-Lerch zeta functions and give applications to the Bernoulli-Nörlund and Apostol-Euler-Nörlund polynomials.
The paper can be summarized as follows. In Section 2, we state our main results. The Section 3 contains the proofs of these results. In Section 4, by means of the main results, we get reduction and its dual formulas for the Bernoulli-Nörlund and Apostol-Euler-Nörlund polynomials.
2 Statement of main results
tends to as . Thus, the lemma is proved. □
Therefore, we obtain the series representation of as follows.
Note that for α be a positive integer N, we have , and by Proposition 2.2, their series representations are given as follows.
We are now able to state our main results.
Theorem 2.4 (Reduction formula)
with for and for .
By dualizing the above theorem, we obtain the following formula.
Theorem 2.5 (Duality formula)
with for and for .
For , we get an extension of Choi’s reduction formula to multiple Hurwitz-Lerch zeta , and we find its dual version.
Substituting in Corollary 2.6, we obtain the following.
where is the Riemann zeta function of order N.
3 Proofs of Theorem 2.4 and Theorem 2.5
Therefore, from equalities (19), (23) and (24), we get our Theorem 2.4.
The proof of Theorem 2.5 is an immediate consequence of the orthogonality properties (5) of the polynomials and , and Theorem 2.4.
4 The Bernoulli-Nörlund and Apostol-Euler-Nörlund polynomials
4.1 Apostol-Euler-Nörlund polynomials
We consider complex numbers α, λ and x such that and . We first prove the analytic continuation of , and we compute special values of .
defines an entire function of .
Remark 4.2 Equality (28) has been proved, using different method, by Luo [, Theorem 2.1].
Now, by applying our Theorem 2.4, Theorem 2.5 and Theorem 4.1, we deduce the reduction and duality formulas for the Apostol-Euler-Nörlund polynomials .
4.2 Explicit formula for the Apostol-Euler-Nörlund polynomials
In particular for , and by using formula (29), we get this explicit formula for the Apostol-Euler-Nörlund polynomials.
4.3 Differential formula for the Apostol-Euler-Nörlund polynomials
4.4 Bernoulli-Nörlund polynomials
The Nörlund polynomials are , see .
Theorem 4.5 ()
defines an entire function of .
For a given non-negative integer (if ), we have a simple pole at with residue .
This proves the analytic continuation of the function as an entire function to the whole complex plane, except simple poles at , if .
This completes the proof of the theorem. □
Remark 4.6 For any positive integer α, the relation (35) recovers the results in the paper .
Hence, from Theorem 2.4 and Theorem 4.1, we deduce the following reduction and duality formulas.
By use of Theorem 4.1 and equalities (36), (37), we get the convolution identities on the Bernoulli-Nörlund polynomials.
Note that from Theorem 4.8 we have the following corollaries.
5 Further applications
We briefly indicate some possible ways to generalize a few known special functions related to the Hurwitz-Lerch zetas functions. We give, in addition, associated reduction and duality formulas.
5.1 Generalized polylogarithms
for if and if .
Therefore, from Theorem 2.4 and Theorem 2.5, the following reduction and duality formulas hold.
5.2 Generalized Fermi-Dirac functions
for , , if , and otherwise.
5.3 Generalized Bose-Einstein functions
The related reduction and duality formulas are then similar to those of the generalized Fermi-Dirac functions.
for , , if , and otherwise.
5.4 Formulas for the generalized Euler-Frobenius polynomials
We consider the Apostol-Euler-Frobenius-Nörlund type polynomials defined as follows.
The so-called Euler-Frobenius polynomials correspond to , and we denote the Apostol-Euler-Frobenius polynomials by .
Dedicated to Professor Hari M Srivastava.
The present investigation was supported by the ‘Equipe Ananlyse et Probabilités’ of the Department of Mathematics at University of Evry.
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