Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus
© Srivastava et al.; licensee Springer 2014
Received: 27 March 2014
Accepted: 4 June 2014
Published: 23 June 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.
Keywordsfractional derivatives generalized Taylor expansion generalized Hurwitz-Lerch zeta functions Riemann zeta function Leibniz rules
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].
it being understood conventionally that and assumed tacitly that the Γ-quotient exists (see, for details, [, p.21 et seq.]).
Several integral representations, relationships with the -function, fractional derivatives, and analytic continuation formulas were established for the function defined in (1.11).
- (i)For , we find that(1.12)
- (iii)Setting and , (1.11) reduces to the function investigated by Goyal and Laddha  as below:(1.14)
- (iv)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)
- (v)A limit case of the generalized Hurwitz-Lerch function , which is of interest in our present investigation, is given by(1.16)
- (vi)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].
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]).
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.
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.
Next, in the year 1971, Osler  obtained the following generalized Taylor-like series expansion involving fractional derivatives.
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.
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.
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 .
provided that both members of (4.1) exist.
Combining (4.3), (4.4), (4.5) with (4.2) and making some elementary simplifications, the asserted result (4.1) follows. □
provided that both members of (4.6) exist.
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 .
provided that both members of (4.11) exist.
for and for z such that .
By combining (4.12) and (4.13), we get the result (4.11) asserted by Theorem 8. □
provided that both sides of (4.14) exist.
Thus, by combining (4.15) and (4.16), we are led to the assertion (4.14) of Theorem 9. □
provided that both sides of (4.17) exist.
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.
provided that both members of (5.1) exist.
Letting in Theorem 7 leads to the following expansion formula.
provided that both members of (5.2) exist.
provided that both sides of (5.3) exist.
Setting in Theorem 10, we obtain the following corollary.
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 .
The authors wish to thank referees for valuable suggestions and comments.
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