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Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus
Advances in Difference Equations volume 2014, Article number: 169 (2014)
Motivated by the recent investigations of several authors, in this paper, we derive several new expansion formulas involving a generalized Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (Integral Transforms Spec. Funct. 22:487-506, 2011). These expansions are obtained by using some fractional calculus theorems such as the generalized Leibniz rules for the fractional derivatives and the Taylor-like expansions in terms of different functions. Several (known or new) special cases are also considered.
MSC:11M25, 11M35, 26A33, 33C05, 33C60.
The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function , the Hurwitz zeta function , and the Lerch zeta function defined by
The Hurwitz-Lerch zeta function defined in (1.5) can be continued meromorphically to the whole complex s-plane, except for a simple pole at with its residue 1. It is well known that
It is worth noting that the Hurwitz-Lerch zeta function defined in (1.5) is also related to several families of special polynomials such as the Bernoulli, the Euler, and the Genocchi polynomials [3–5].
Here, and for the remainder of this paper, denotes the Pochhammer symbol defined, in terms of the gamma function, by
it being understood conventionally that and assumed tacitly that the Γ-quotient exists (see, for details, [, p.21 et seq.]).
Clearly, we find from the definition (1.6) that
Several integral representations, relationships with the -function, fractional derivatives, and analytic continuation formulas were established for the function defined in (1.11).
It is worth noting the following special or limit cases of the function .
For , we find that(1.12)
in terms of the generalized Hurwitz-Lerch zeta function defined in (1.6).
Setting and , (1.11) reduces to the function investigated by Goyal and Laddha  as below:(1.14)
In (1.11), we put and . Then, by the familiar principle of confluence, the limit case when , would yield the Mittag-Leffler type function studied by Barnes , namely(1.15)
A limit case of the generalized Hurwitz-Lerch function , which is of interest in our present investigation, is given by(1.16)
Another limit case of the generalized Hurwitz-Lerch function is given by(1.17)
which, for , reduces at once to the function defined by (1.9).
It is fairly straightforward to see that if we let in (1.18), then we obtain the generalized Hurwitz-Lerch zeta function .
The aim of this paper is to extend several interesting results obtained recently by Gaboury and Bayad  and by Gaboury  to the Hurwitz-Lerch zeta function introduced and studied by Srivastava et al. . This paper is organized as follows. Section 2 is devoted to the representation of the fractional derivatives based on Pochhammer’s contour of integration. In Section 3, we recall some major fractional calculus theorems, that is, two generalized Leibniz rules and three Taylor-like expansions. Section 4 is dedicated to the proofs of the main results and, finally, Section 5 aims to provide some (new or known) special cases.
2 Pochhammer contour integral representation for fractional derivative
The fractional derivative of arbitrary order α, , is an extension of the familiar n th derivative of the function with respect to to non-integral values of n and denoted by . The aim of this concept is to generalize classical results of the n th order derivative to fractional order. Most of the properties of the classical calculus have been expanded to fractional calculus. For instance, the composition rule, the Leibniz rule, the chain rule and the Taylor and Laurent series. Fractional calculus provides tools that make easier to deal with special functions of mathematical physics. Many examples of the use of fractional derivatives appear in the literature: ordinary and partial differential equations, integral equations, integro-differential equations of non-integer order. Many other applications have been investigated through various field of science and engineering. For more details on fractional calculus, the reader could read [18–21].
where the integration is carried out along a straight line from 0 to z in the complex ξ-plane. By integrating by part m times, we obtain
This allows us to modify the restriction to (see ).
Another representation for the fractional derivative is based on the Cauchy integral formula. This representation, too, has been widely used in many interesting papers (see, for example, the work of Osler [25–28]).
Definition 1 Let be analytic in a simply connected region ℛ of the complex z-plane. Let be regular and univalent on ℛ and let be an interior point of ℛ. Then, if α is not a negative integer, p is not an integer, and z is in , we define the fractional derivative of order α of with respect to by
For non-integers α and p, the functions and in the integrand have two branch lines which begin, respectively, at and , and both branches pass through the point without crossing the Pochhammer contour at any other point as shown in Figure 1. Here denotes the principal value of the integrand in (2.3) at the beginning and the ending point of the Pochhammer contour which is closed on the Riemann surface of the multiple-valued function .
Remark 1 In Definition 1, the function must be analytic at . However, it is interesting to note here that, if we could also allow to have an essential singularity at , then (2.3) would still be valid.
Remark 2 In case the Pochhammer contour never crosses the singularities at and in (2.3), then we know that the integral is analytic for all p and for all α and for z in . Indeed, in this case, the only possible singularities of are and , which can directly be identified from the coefficient of the integral (2.3). However, by integrating by parts N times the integral in (2.3) by two different ways, we can show that and are removable singularities (see, for details, ).
Adopting the Pochhammer-based representation for the fractional derivative modifies the restriction to the case when p not a negative integer.
These last restrictions become not a negative integer and by making use of the Pochhammer-based representation for the fractional derivative.
The fractional derivative formula (2.5) can be specialized to deduce other results. As example, upon setting in (2.5), we obtain
Another fractional derivative formula that will be very useful in the present investigation is given by the next formula:
where the Hurwitz-Lerch zeta function occurring in (2.7) is a specialized case of the multiparameters extension of the generalized Hurwitz-Lerch zeta function defined in (1.18).
3 Important results involving fractional calculus
In this section, we recall five very important theorems related to fractional calculus that will play central roles in our work. Each of these theorems is the generalized Leibniz rules for fractional derivatives and the Taylor-like expansions in terms of different types of functions.
First of all, we give two generalized Leibniz rules for fractional derivatives. Theorem 1 is a slightly modified theorem obtained in 1970 by Osler . Theorem 2 was given, some years ago, by Tremblay et al.  with the help of the properties of Pochhammer’s contour representation for fractional derivatives.
Theorem 1 (i) Let ℛ be a simply connected region containing the origin. (ii) Let and satisfy the conditions of Definition 1 for the existence of the fractional derivative. Then, for and , the following Leibniz rule holds true:
Theorem 2 (i) Let ℛ be a simply connected region containing the origin. (ii) Let and satisfy the conditions of Definition 1 for the existence of the fractional derivative. (iii) Let be the region of analyticity of the function and be the region of analyticity of the function . Then, for
the following product rule holds true:
Next, in the year 1971, Osler  obtained the following generalized Taylor-like series expansion involving fractional derivatives.
Theorem 3 Let be an analytic function in a simply connected region ℛ. Let α and γ be arbitrary complex numbers and
with a regular and univalent function without any zero in ℛ. Let a be a positive real number and
Let b and be two points in ℛ such that and let
Then the following relationship holds true:
In particular, if and , then and (3.3) reduces to the following form:
We next recall that Tremblay et al.  discovered the power series of an analytic function in terms of the rational expression , where and are two arbitrary points inside the region ℛ of analyticity of . In particular, they obtained the following result.
Theorem 4 (i) Let c be real and positive and let
(ii) Let be analytic in the simply connected region ℛ with and being interior points of ℛ. (iii) Let the set of curves
be defined by
which are the Bernoulli type lemniscates (see Figure 2) with center located at and with double-loops in which one loop leads around the focus point
and the other loop encircles the focus point
for each t such that . (iv) Let
denote the principal branch of that function which is continuous and inside , cut by the respective two branch lines defined by
such that is real when . (v) Let satisfy the conditions of Definition 1 for the existence of the fractional derivative of of order α for , denoted by , where α and p are real or complex numbers. (vi) Let
Then, for arbitrary complex numbers μ, ν, γ, and for z on defined by
The case of Theorem 4 reduces to the following form:
Tremblay and Fugère  developed the power series of an analytic function in terms of the function , where and are two arbitrary points inside the analyticity region ℛ of . Explicitly, they gave the following theorem.
Theorem 5 Under the assumptions of Theorem 4, the following expansion formula holds true:
As a special case, if we set , (), and in (3.11), we obtain
4 A set of main results for the generalized Hurwitz-Lerch zeta function
In this section, we present the new expansion formulas involving the generalized Hurwitz-Lerch zeta functions .
Theorem 6 Under the assumptions of Theorem 1, the following expansion holds true:
provided that both members of (4.1) exist.
Proof Setting and in Theorem 1 with and , we obtain
which, with the help of (2.4) and (2.5), yields
Combining (4.3), (4.4), (4.5) with (4.2) and making some elementary simplifications, the asserted result (4.1) follows. □
Theorem 7 Under the hypotheses of Theorem 2, the following expansion formula holds true:
provided that both members of (4.6) exist.
Proof Upon first substituting and in Theorem 2 and then setting
in which both and satisfy the conditions of Theorem 2, we have
Now, by using (2.4) and (2.5), we find that
Thus, finally, the result (4.6) follows by combining (4.8), (4.9), (4.10), and (4.7). □
We now shift our focus on the different Taylor-like expansions in terms of different types of functions involving the generalized Hurwitz-Lerch zeta functions .
Theorem 8 Under the assumptions of Theorem 3, the following expansion formula holds true:
provided that both members of (4.11) exist.
Proof Setting in Theorem 3 with , , and , we have
for and for z such that .
Now, by making use of (2.7) with and , we find that
By combining (4.12) and (4.13), we get the result (4.11) asserted by Theorem 8. □
Theorem 9 Under the hypotheses of Theorem 4, the following expansion formula holds true:
for and for z on defined by
provided that both sides of (4.14) exist.
Proof By taking in Theorem 4 with , , , and , we find that
Now, with the help of the relation (2.7) with and , we have
Thus, by combining (4.15) and (4.16), we are led to the assertion (4.14) of Theorem 9. □
Theorem 10 Under the hypotheses of Theorem 5, the following expansion formula holds true:
for and for z on defined by
provided that both sides of (4.17) exist.
Proof Putting in Theorem 5 with , , , and , we find that
With the help of relation (2.7), we have
Thus, by combining (4.18) and (4.19), we obtain the desired result (4.17). □
5 Corollaries and consequences
This section is devoted to the presentation of some special cases of the main results. These special cases and consequences are given in the form of the following corollaries.
Setting in Theorem 6 with , dividing by and taking the limit when , we deduce the following expansion formula.
Corollary 1 Under the hypotheses of Theorem 6, the following expansion holds true:
provided that both members of (5.1) exist.
Letting in Theorem 7 leads to the following expansion formula.
Corollary 2 Under the assumptions of Theorem 7, the following expansion formula holds true:
provided that both members of (5.2) exist.
Corollary 3 Under the hypotheses of Theorem 9, the following expansion formula holds true:
for z on defined by
provided that both sides of (5.3) exist.
Setting in Theorem 10, we obtain the following corollary.
Corollary 4 Under the hypotheses of Theorem 10, the following expansion holds true:
for and for z on defined by
provided that both sides of (5.5) exist.
In our series of forthcoming papers, we propose to consider and investigate analogous expansion formulas and other results involving the more general multi-parameter family of the Hurwitz-Lerch zeta function (1.18) and also their λ-extensions considered recently by Srivastava et al.  and Srivastava .
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The authors wish to thank referees for valuable suggestions and comments.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Srivastava, H.M., Gaboury, S. & Bayad, A. Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus. Adv Differ Equ 2014, 169 (2014). https://doi.org/10.1186/1687-1847-2014-169
- fractional derivatives
- generalized Taylor expansion
- generalized Hurwitz-Lerch zeta functions
- Riemann zeta function
- Leibniz rules