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International Journal of Analysis

Volume 2014, Article ID 680850, 14 pages

http://dx.doi.org/10.1155/2014/680850
Research Article

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

1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4

2Department of Mathematics and Computer Science, University of Quebec at Chicoutimi, Chicoutimi, QC, Canada G7H 2B1

Received 28 February 2014; Accepted 16 April 2014; Published 13 May 2014

Academic Editor: Shamsul Qamar

Copyright © 2014 H. M. Srivastava et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

The Hurwitz-Lerch zeta function 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 et seq.]) 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 respectively.

The Hurwitz-Lerch zeta function is connected with other special functions of analytic number theory such as the polylogarithmic function (or de Jonquière’s function) : and the Lipschitz-Lerch zeta function (see [1, page 122, Equation 2.5   ]) The Hurwitz-Lerch zeta function defined in (7) can be continued meromorphically to the whole complex -plane, except for a simple pole at with its residue 1. It is well known that

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 defined by (7) (see also [9]). In particular, they considered the following functions: with Here, and for the remainder of this paper, denotes the Pochhammer symbol defined, in terms of the gamma function, by it is being understood conventionally that and assumed tacitly that the -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 :

provided that both sides of (11) exist.

Definition 1. The involved in the right-hand side of (11) is the generalized Fox’s -function introduced by Inayat-Hussain [11, page 4126] Here the parameters and the exponents can take noninteger values and is a Mellin-Barnes type contour starting at the point and terminating at the point with the usual indentations to separate one set of poles from the other set of poles.

Buschman and Srivastava [12, page 4708] established that the sufficient conditions for the absolute convergence of the contour integral in (12) are given by 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 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.

2. 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 [1820]); that is, where the integration is carried out along a straight line from to in the complex -plane. By integrating by part times, we obtain This allows us to modify the restriction to (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 [2124]).

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].

Definition 2. Let be analytic in a simply connected region of the complex -plane. Let be regular and univalent on and let be an interior point of . Then, if is not a negative integer, is not an integer, and is in , we define the fractional derivative of order of with respect to by For nonintegers and , 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 (20) at the beginning and the ending point of the Pochhammer contour which is closed on the Riemann surface of the multiple-valued function (see Figure 2).

680850.fig.001
Figure 1: Pochhammer’s contour.
680850.fig.002
Figure 2: Multiloops contour.

Remark 3. In Definition 2, 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 (20) would still be valid.

Remark 4. In case the Pochhammer contour never crosses the singularities at and in (20), then we know that the integral is analytic for all and for all and for in . Indeed, in this case, the only possible singularities of are and , which can directly be identified from the coefficient of the integral (20). However, by integrating by parts times the integral in (20) by two different ways, we can show that and are removable singularities (see, for details, [25]).

In their work, Srivastava et al. [7] made use of the following fractional calculus result obtained by Srivastava et al. [27, page 97, Equation ]:

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 ,not a negative integer, and .

The parameters involved in the fractional derivative formula (22) can be specialized to deduce other results. For example, setting in (22) and making the following substitutions , , , , , and lead to Furthermore, if we put in (23), then we obtain 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: This last result can be established with the help of the following well known formula [28, page 83, Equation ]: Adopting the Pochhammer based representation for the fractional derivative modifies the restriction to the case when is not a negative integer.

3. 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 be a simply connected region containing the origin. (ii) Let and satisfy the conditions of Definition 2 for the existence of the fractional derivative. Then, for and , the following Leibniz rule holds true:

Theorem 6. (i) Let be a simply connected region containing the origin. (ii) Let and satisfy the conditions of Definition 2 for the existence of the fractional derivative. (iii) Let be the region of analyticity of the function and let be the region of analyticity of the function . 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 . Let and be arbitrary complex numbers and with a regular and univalent function without any zero in . Let be a positive real number and Let and be two points in such that and let Then the following relationship holds true:

In particular, if and , then 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, 3134].

We next recall that Tremblay et al. [35] obtained the power series of an analytic function in terms of the rational expression , where and are two arbitrary points inside the region of analyticity of . In particular, they obtained the following result.

Theorem 8. (i) Let be real and positive and let

(ii) Let be analytic in the simply connected region with and being interior points of . (iii) Let the set of curves be defined by where which are the Bernoulli type lemniscates 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 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 2 for the existence of the fractional derivative of of order for , denoted by , where and are real or complex numbers. (vi) Let Then, for arbitrary complex numbers , , and for on defined by where

The case of Theorem 8 reduces to the following form:

Tremblay and Fugère [36] developed the power series of an analytic function in terms of the function , where and are two arbitrary points inside the analyticity region of . Explicitly, they showed the following theorem.

Theorem 9. Under the assumptions of Theorem 8, the following expansion formula holds true: where

As special case, if we set , , and in (48), we obtain

Finally, Osler [21, page 290, Theorem 2] discovered a fundamental relation from which he deduced the generalized chain rule for the fractional derivatives. This result is recalled here as Theorem 10.

Theorem 10. Let and be defined and analytic in the simply connected region of the complex -plane and let the origin be an interior or boundary point of . Suppose also that and are regular univalent functions on and that . Let vanish over simple closed contour in through the origin. Then the following relation holds true:

The relation (51) allows us to obtain very easily known and new summation formulas involving special functions of mathematical physics.

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

4. Relations Involving the Extended Multiparameters Hurwitz-Lerch Zeta Function

In this section, we present the new expansion formulas involving the extended multiparameters Hurwitz-Lerch zeta function .

Theorem 12. Under the hypotheses of Theorem 5, the following expansion formula holds true: provided that both members of (54) exist.

Proof. Setting and in Theorem 5 with and , we obtain which, with the help of (22), (26), and (27), yields Combining (56) with (55) and making some elementary simplifications, the asserted result (54) follows.

Theorem 13. Under the hypotheses of Theorem 6, the following expansion formula holds true: provided that both members of (57) exist.

Proof. Upon first substituting and in Theorem 6 and then setting in which both and satisfy the conditions of Theorem 6, we have Now, by using (22), (26), and (27), we find that Thus, finally, the result (57) follows by combining (60) and (59).

We now shift our focus to the different Taylor-like expansions in terms of different types of functions involving the extended multiparameters Hurwitz-Lerch zeta functions .

Theorem 14. Under the assumptions of Theorem 7, the following expansion formula holds true: provided that both members of (61) exist.

Proof. Setting in Theorem 7 with , , and , we have for and for such that .

Now, by making use of (26) with and , we find that By combining (62) and (63), we get result (61) asserted by Theorem 14.

Theorem 15. Under the hypotheses of Theorem 8, the following expansion formula holds true: for and for on defined by provided that both sides of (64) exist.

Proof. By taking in Theorem 8 with , , , and , we find that Now, with the help of relation (26) with and , we have Thus, by combining (66) and (67), we are led to assertion (64) of Theorem 15.

Theorem 16. Under the hypotheses of Theorem 9, the following expansion formula holds true: for and for on defined by provided that both sides of (68) exist.

Proof. Setting in Theorem 9 with , , , and , we find that With the help of the relation in (26), we have