Reduction and duality of the generalized Hurwitz-Lerch zetas
Fixed Point Theory and Applications volume 2013, Article number: 82 (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
We use the falling factorial , the binomial notation and the polynomials and , which are defined by equalities (1) and (2). For a non-negative integer n, parameter a and an indeterminate X, we have
where , and and are the Stirling numbers of the first and the second kind, respectively. Moreover, these polynomials satisfy the orthogonality formulas
where is the Kronecker delta symbol.
For a given positive integer N and complex with and , the multiple Hurwitz-Lerch zeta function of order N is defined by the series
for a complex s such that if and if , and its integral representation is given by
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.
The multiple Hurwitz zeta function of order N corresponds to the multiple Hurwitz-Lerch zeta function , while the Hurwitz zeta function is simply . It is shown in  that the multiple Hurwitz zeta function can be reduced to a finite sum of the Hurwitz zeta functions with Stirling numbers in coefficients. Precisely, we have
For more details about the formula (8), see . From the analytic continuations of and to the whole complex plane, the generalized n th-Bernoulli polynomial of order N and the n th-Bernoulli polynomial are related to and by the formulas
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
Let us consider complex numbers α, λ and x such that and . We define the generalized Hurwitz-Lerch zeta function by the integral representation
Lemma 2.1 Let τ be a positive real number, and let α, λ be complex numbers such that . Then the series of functions
is absolutely and uniformly convergent on and
Proof Indeed, we have for all positive integer k the majoration
and the ratio
tends to as . Thus, the lemma is proved. □
On the other hand, the integral
is absolutely convergent for for , and for . Therefore, by means of Lemma 2.1, we can interchange the summation and integration to obtain
Therefore, we obtain the series representation of as follows.
Proposition 2.2 For any complex numbers α, λ and x such that and , we have
Note that for α be a positive integer N, we have , and by Proposition 2.2, their series representations are given as follows.
Corollary 2.3 For any positive integer N, complex number λ and x such that and , we have
We are now able to state our main results.
Theorem 2.4 (Reduction formula)
For any non-negative integer N and complex numbers α, λ, x such that and , the following reduction formula holds:
with for and for .
By dualizing the above theorem, we obtain the following formula.
Theorem 2.5 (Duality formula)
For any non-negative integer N and complex numbers α, λ, x such that and , we have
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.
Corollary 2.6 Let N be a positive integer, and let x, λ be complex numbers such that , . Then, for any complex s with , we have the reduction and duality formulas
Substituting in Corollary 2.6, we obtain the following.
Corollary 2.7 Let N be a positive integer. For any complex s with , we obtain the formula
and its dual
where is the Riemann zeta function of order N.
3 Proofs of Theorem 2.4 and Theorem 2.5
In the sequel, all the parameters under consideration are subject to the conditions of the theorem. For fixed parameters α, x and λ, the function is smooth and satisfies the differential identity
From the identity (18), with the help of integration by parts, we get
The hypotheses of the theorems ensure that , and hence we have
where the functions are the translation operators. By switching α in the identity (19), we see that the N equalities
hold. The composition of these equalities yields the relation
On the other hand, we have
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
The Apostol-Euler-Nörlund polynomials are defined, for , by the generating function
We consider complex numbers α, λ and x such that and . We first prove the analytic continuation of , and we compute special values of .
Theorem 4.1 Let λ be a complex number with . The function has analytic continuation to the whole complex plane, except possible simple poles at non-positive integers with residue
Moreover, the function has analytic continuation as an entire function to the whole complex plane, and for all non-negative integer n, we have
Proof Let λ be a complex number other than 1. We choose a real number so that . We split the integral from zeta’s definition as
As and , for all complex number α, the integral
defines an entire function of .
We substitute in the first integral, for , the generating expansion of the Apostol-Euler-Nörlund polynomials
It hence follows that
For a given non-negative integer m, we know that
and . This proves the analytic continuation of the function as an entire function to the whole complex plane, and
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 .
Theorem 4.3 For any non-negative integers n, N and complex numbers α, with , we have
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.
Proposition 4.4 Let α, λ be complex numbers with , . For any positive integer N, we have
The above formula, when combined with the well-known equality
gives this other explicit expression
4.3 Differential formula for the Apostol-Euler-Nörlund polynomials
We consider the differential operator . From the series representation (2.2) of the Hurwitz-Lerch zeta functions, we have
Using Theorem 4.1 at , we obtain the differential formula
4.4 Bernoulli-Nörlund polynomials
In this subsection we investigate convolution formulas for the Bernoulli-Nörlund polynomials . We define them by the generating function
The Nörlund polynomials are , see .
We set , and we introduce the modified Hurwitz-Lerch zeta function defined by the integral representation
Theorem 4.5 ()
Let α, x be complex numbers with . The function has analytic continuation to the whole complex plane, except simple poles at with residue , for any non-negative integer (when ). Moreover, for all non-negative integer , we have
The proof is similar to that of Theorem 4.1. We split the integral
As , for all complex number α, the integral
defines an entire function of .
We use in the first integral, for , the generating function of the Bernoulli-Nörlund polynomials
It hence follows that
For a given non-negative integer (if ), we have a simple pole at with residue .
For a given integer , it is known that
and . Then we obtain
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.
Theorem 4.7 Under the hypothesis of Theorem 2.4, we have
By use of Theorem 4.1 and equalities (36), (37), we get the convolution identities on the Bernoulli-Nörlund polynomials.
Theorem 4.8 For any non-negative integer n and any positive integer N, we have the convolution identity and its dual version
Note that from Theorem 4.8 we have the following corollaries.
Corollary 4.9 Under the hypothesis of Theorem 4.8, for , and indeterminate α, we obtain, among the so-called Nörlund polynomials , the formula
Corollary 4.10 Under the hypothesis of Theorem 4.8, for , we obtain the following formula:
Corollary 4.11 Under the hypothesis of Theorem 4.8, and if , , we obtain Euler’s identity type on the Bernoulli numbers of order N
This formula is an analogue of the nice identity for Bernoulli numbers obtained by Euler and given by
Corollary 4.12 Under the hypothesis of Theorem 4.8, taking in the dual formula (39), we obtain
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
Let α, λ be complex numbers such that and . We define the generalized polylogarithms by the equality
which is equivalent to the equalities
for if and if .
The ordinary polylogarithm corresponds to
Therefore, from Theorem 2.4 and Theorem 2.5, the following reduction and duality formulas hold.
Theorem 5.1 Under the hypothesis of Theorem 2.4 on the parameters α, λ, N and s, we have the reduction and duality relations for the generalized polylogarithms
5.2 Generalized Fermi-Dirac functions
Following Srivastava et al. , we extend the definition of the Fermi-Dirac functions, with parameter α with , as follows:
and if and if . Alternatively, they have a series representation related to the Hurwitz-Lerch zetas. Under the same conditions on parameters as above, we have
The ordinary Fermi-Dirac function is given by
Theorem 5.2 We have the reduction and duality relations for the generalized Fermi-Dirac functions
for , , if , and otherwise.
5.3 Generalized Bose-Einstein functions
The generalized Bose-Einstein functions can also be defined as follows. As the case in , we can define them by their integral representations
and if , and otherwise. Alternatively, their series representation and relationship with Hurwitz-Lerch zetas are given by the equalities
The ordinary Bose-Einstein function corresponds to
The related reduction and duality formulas are then similar to those of the generalized Fermi-Dirac functions.
Theorem 5.3 We have the reduction and duality relations for the generalized Bose-Einstein 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.
For and , the Frobenius-Euler-Nörlund polynomials are defined through the generating function
The so-called Euler-Frobenius polynomials correspond to , and we denote the Apostol-Euler-Frobenius polynomials by .
we easily see that for all non-negative integer n, we have
and using the explicit formula (30) of the Apostol-Euler-Nörlund polynomials, we get that of the Apostol-Euler-Frobenius-Nörlund polynomials
On the other hand, using equality (33), we deduce the differential formula for the Apostol-Euler-Frobenius-Nörlund polynomials
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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.
The authors declare that they have no competing interests.
All authors completed the paper, read and approved the final manuscript.
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Bayad, A., Chikhi, J. Reduction and duality of the generalized Hurwitz-Lerch zetas. Fixed Point Theory Appl 2013, 82 (2013). https://doi.org/10.1186/1687-1812-2013-82