New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications

We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (2011). These expansions are obtained by using some fractional calculus methods such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also given.


Introduction
The Hurwitz-Lerch zeta function Φ( , , ) which is one of the fundamentally important higher transcendental functions is defined by (see, e.g., [1, page 121 et seq.]; see also [2] and [3, page 194 The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function ( ), the Hurwitz zeta function The Hurwitz-Lerch zeta function Φ( , , ) defined in (7) can be continued meromorphically to the whole complex -plane, Motivated by the works of Goyal and Laddha [4], Lin and Srivastava [5], Garg et al. [6], and other authors, Srivastava et al. [7] (see also [8]) investigated various properties of a natural multiparameter extension and generalization of the Hurwitz-Lerch zeta function Φ( , , ) defined by (7) (see also [9]). In particular, they considered the following functions: Here, and for the remainder of this paper, ( ) denotes the Pochhammer symbol defined, in terms of the gamma function, by (10) it is being understood conventionally that (0) 0 := 1 and assumed tacitly that the Γ-quotient exists (see, for details, [10, page 21 et seq.]).
In their work, Srivastava et al. [7, page 504, Theorem 8] also proved the following relation for the function Φ provided that both sides of (11) exist.
Buschman and Srivastava [12, page 4708] established that the sufficient conditions for the absolute convergence of the contour integral in (12) are given by > 0 (15) and the region of absolute convergence is Note that when the -function reduces to the well-known Fox's -function (see [13]). This paper is devoted to extending several interesting results obtained recently by Srivastava et al. [14] (see also [15,16]) to the extended multiparameter Hurwitz-Lerch zeta function Φ ( 1 ,..., , 1 ,..., ) 1 ,..., ; 1 ,..., ( , , ) introduced and studied by Srivastava et al. [7]. In Section 2, we give the representation of the fractional derivatives based on the Pochhammer's contour of integration. Section 3 aims at recalling some major fractional calculus theorems, that is, two generalized Leibniz rules and three Taylor-like expansions as well as a fundamental relation linked to the generalized chain rule for the fractional derivatives. In the two remaining sections, we, respectively, present and prove the main results of this paper and we give some special cases.

Pochhammer Contour Integral Representation for Fractional Derivative
The most familiar representation for the fractional derivative of order of ( ) is the Riemann-Liouville integral [17] (see also [18][19][20]); that is, where the integration is carried out along a straight line from 0 to in the complex -plane. By integrating by part times, we obtain Branch line for Branch line for This allows us to modify the restriction R( ) < 0 to R( ) < (see [20]). 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 works of Osler [21][22][23][24]).
The relatively less restrictive representation of the fractional derivative according to parameters appears to be the one based on the Pochhammer's contour integral introduced by Lavoie et al. [25] and Tremblay [26].  we could also allow ( ) to have an essential singularity at = −1 (0), then (20) would still be valid.
In their work, Srivastava et al. [7] in order to derive the following important fractional derivative formula for this work: This fractional calculus formula was obtained by using the Riemann-Liouville representation for the fractional derivative. Adopting the Pochhammer based representation for the fractional derivative, these last restrictions become + ] − 1 ,not a negative integer, and > 0.
The parameters involved in the fractional derivative formula (22) can be specialized to deduce other results. For example, setting − 1 = = 1 in (22) and making the following substitutions 1 International Journal of Analysis 5 Furthermore, if we put = = = 1 in (23), then we obtain (] not a negative integer; > 0) .
Finally, letting = ] in (24), this yields after elementary calculations Another fractional derivative formula that will be very useful in this work is given by the following formula: Adopting the Pochhammer based representation for the fractional derivative modifies the restriction to the case when is not a negative integer.

Some Fundamental Theorems Involving Fractional Calculus
In this section, we recall six fundamental 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, the Taylor-like expansions in terms of different types of functions, and a fundamental formula related to the generalized chain rule for fractional derivatives.
First of all, we give two generalized Leibniz rules for fractional derivatives. Theorem 5 is a slightly modified theorem obtained in 1970 by Osler [22]. Theorem 6 was given, some years ago, by Tremblay et al. [29] with the help of the properties of Pochhammer's contour representation for fractional derivatives.
Theorem 5. (i) Let R be a simply connected region containing the origin. (ii) Let ( ) and V( ) satisfy the conditions of Definition 2 for the existence of the fractional derivative. Then, for R( + ) > −1 and ∈ C, the following Leibniz rule holds true: Then, for the following product rule holds true: Next, in 1971, Osler [30] established the following generalized Taylor-like series expansion involving fractional derivatives.

Theorem 7. Let ( ) be an analytic function in a simply connected region R. Let and be arbitrary complex numbers and
Let and 0 be two points in R such that ̸ = 0 and let 6 International Journal of Analysis Then the following relationship holds true: In particular, if 0 < ≦ 1 and ( ) = ( − 0 ), then = 0 and the formula (34) reduces to the following form: This last formula (35) is usually referred to as the Taylor-Riemann formula and has been studied in several papers [23,[31][32][33][34]. We next recall that Tremblay et al. [35] obtained the power series of an analytic function ( ) in terms of the rational expression (( − 1 )/( − 2 )), where 1 and 2 are two arbitrary points inside the region R of analyticity of ( ). In particular, they obtained the following result.
be defined by where which are the Bernoulli type lemniscates with center located at ( 1 + 2 )/2 and with double-loops in which one loop 1 ( ) leads around the focus point and the other loop 2 ( ) encircles the focus point for each such that 0 < ≦ . (iv) Let denote the principal branch of that function which is continuous and inside ( ), cut by the respective two branch lines ± defined by Then, for arbitrary complex numbers , ], and for on 1 (1) defined by where The case 0 < ≦ 1 of Theorem 8 reduces to the following form: International Journal of Analysis 7 Tremblay and Fugère [36] developed the power series of an analytic function ( ) in terms of the function ( − 1 )( − 2 ), where 1 and 2 are two arbitrary points inside the analyticity region R of ( ). Explicitly, they showed the following theorem.
By applying relation (51), Gaboury and Tremblay [37] proved the following corollary which will be useful in the next section.

Corollary 11. Under the hypotheses of Theorem 10, let be a positive integer. Then the following relation holds true:
where ( )

Relations Involving the Extended Multiparameters Hurwitz-Lerch Zeta
provided that both members of (61) exist.
International Journal of Analysis for 0 ̸ = 0 and for such that | − 0 | = | 0 |. Now, by making use of (26) with = 0 and = , we find that  for > 0 and for on 1 (1) defined by provided that both sides of (64) exist.
Proof. Putting = and letting ( ) = Φ With the help of the definition of given by (53), we find for the left-hand side of (74) that We now expand each factor in the product in (74) in power series and replace extended multiparameters Hurwitz-Lerch zeta function by its series representation. We thus find for the right-hand side of (74) that Finally, by combining (75) and (76), we obtain the result (72) asserted by Theorem 17.
Remark 18. Each of the previous theorems can be written in terms of -function given in Definition 1. For instance, if we make use of (11), then Theorem 17 becomes

Corollaries and Consequences
We conclude this paper by presenting some special cases of the main results. These special cases and consequences are given in the form of the following corollaries. Setting = 3 in Theorem 17, we obtain the following corollary.
provided that both sides of (78) exist.

International Journal of Analysis
for such that | − 0 | = | 0 | and provided that both members of (80) exist.
for on 1 (1) defined by provided that both sides of (86) exist.
The function Ψ * involved in the left-hand side of (86) is the Fox-Wright function ( , ∈ N 0 ) with numerator parameters 1 , . . . , and denominator parameters 1 , . . . , such that where the equality in the convergence condition holds true for suitably bounded values of | | given by ) .
In our series of forthcoming papers, we propose to consider and investigate analogous expansion formulas and other results involving the -extensions of the generalized Hurwitz-Lerch zeta functions. These functions have been investigated recently by Srivastava et al. [41] (see also [42]).