International Journal of Analysis

Volume 2014 (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

H. M. Srivastava,¹ Sébastien Gaboury,² and Richard Tremblay²

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 [18–20]); 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 [21–24]).

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].

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).

Figure 1: Pochhammer’s contour.

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 andin (20), then we know that the integral is analytic for alland for alland forin . Indeed, in this case, the only possible singularities ofareand, which can directly be identified from the coefficient of the integral (20). However, by integrating by partstimes the integral in (20) by two different ways, we can show thatandare 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, settingin (22) and making the following substitutions,,,,, andlead to

Furthermore, if we putin (23), then we obtain

Finally, lettingin (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 whenis 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 andsatisfy 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 andsatisfy the conditions of Definition 2 for the existence of the fractional derivative. (iii) Let be the region of analyticity of the functionand 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. Letbe an analytic function in a simply connected region . Letandbe arbitrary complex numbers and

witha regular and univalent function without any zero in . Letbe a positive real number and

Let and be two points in such thatand 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, 31–34].

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) Letbe real and positive and let

(ii) Letbe analytic in the simply connected region withandbeing 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 loopencircles the focus point

for eachsuch 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) Letsatisfy the conditions of Definition 2 for the existence of the fractional derivative ofof orderfor , denoted by, whereandare real or complex numbers. (vi) Let

Then, for arbitrary complex numbers , , and for on defined by

where

The caseof Theorem 8 reduces to the following form:

Tremblay and Fugère [36] developed the power series of an analytic functionin terms of the function, whereandare 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, , andin (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. Letandbe 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 thatandare regular univalent functions on and that. Letvanish 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, letbe 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. Settingandin Theorem 5 withand, 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 substitutingandin Theorem 6 and then setting

in which bothandsatisfy 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. Settingin Theorem 7 with,, and, we have

forand forsuch that.

Now, by making use of (26) withand, 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:

forand forondefined by

provided that both sides of (64) exist.

Proof. By takingin Theorem 8 with,,, and, we find that

Now, with the help of relation (26) withand, 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:

forand forondefined by

provided that both sides of (68) exist.

Proof. Settingin Theorem 9 with,, , and, we find that

With the help of the relation in (26), we have

Thus, by combining (70) and (71), we obtain desired result (68).

Finally, from Corollary 11 given in the preceding section, we obtain the following new relation involving the extended multiparameters Hurwitz-Lerch zeta function.

Theorem 17. Under the hypotheses of Corollary 11, letbe a positive integer. Then the following relation holds true:

where and denotes the Lauricella function of variables defined by [38, page 60]

provided that both sides of (72) exist.

Proof. Puttingand lettingin Corollary 11, we get

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

5. 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.

Settingin Theorem 17, we obtain the following corollary.

Corollary 19. Under the hypotheses of Theorem 17, the following expansion formula holds true:

wheredenotes the first Appell function defined by [38, page 22]

provided that both sides of (78) exist.

Setting,, and making the following substitutions,,,,, andin Theorem 14 lead to the following expansion formula given recently by Srivastava et al. [14].

Corollary 20. Under the hypotheses of Theorem 14, the following expansion formula holds true

for such thatand provided that both members of (80) exist.

Remark 21. The function, which occurs in Corollary 20 above, as well as its multiparameter generalizations, was introduced and investigated in a series of papers by Srivastava et al. (see [7, page 491, Equation ]; see also [8, 9, 39]) and is defined as follows:

Putting,, and setting,, and in Theorem 15 reduce to the following expansion formula given recently by Gaboury [15, Equation ].

Corollary 22. Under the hypotheses of Theorem 15, the following expansion formula holds true:

forondefined by

provided that both sides of (82) exist.

Remark 23. The function, which occurs in Corollary 22 above, was introduced and studied by Goyal and Laddha [4, page 100, Equation ] and is defined as follows:

Setting,,, andwith

in Theorem 16, we deduce the following expansion formula.

Corollary 24. Under the hypotheses of Theorem 16, the following formula holds true:

forondefined by

provided that both sides of (86) exist.

The function involved in the left-hand side of (86) is the Fox-Wright functionwithnumerator parametersand denominator parameterssuch that

defined by (see, for details, [17, 40])

where the equality in the convergence condition holds true for suitably bounded values ofgiven 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]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

H. M. Srivastava, “Some formulas for the Bernoulli and Euler polynomials at rational arguments,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 129, no. 1, pp. 77–84, 2000.

H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, The Netherlands, 2012.

S. P. Goyal and R. K. Laddha, “On the generalized Riemann zeta functions and the generalized Lambert transform,” Ganita Sandesh, vol. 11, no. 2, pp. 99–108, 1997.

S.-D. Lin and H. M. Srivastava, “Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 725–733, 2004.

M. Carg, K. Jain, and S. L. Kalla, “A further study of general Hurwitz-Lerch zeta function,” Algebras, Groups and Geometries, vol. 25, no. 3, pp. 311–319, 2008.

H. M. Srivastava, R. K. Saxena, T. K. Pogány, and R. Saxena, “Integral and computational representations of the extended Hurwitz-Lerch zeta function,” Integral Transforms and Special Functions, vol. 22, no. 7, pp. 487–506, 2011.

H. M. Srivastava, “Some families of mixed multilateral generating relations,” Journal of Rajasthan Academy of Physical Sciences, vol. 1, no. 1, pp. 1–6, 2002.

H. M. Srivastava, “Some generalizations and basic (or -) extensions of the Bernoulli, Euler and Genocchi polynomials,” Applied Mathematics & Information Sciences, vol. 5, no. 3, pp. 390–444, 2011.

H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1984.

A. A. Inayat-Hussain, “New properties of hypergeometric series derivable from Feynman integrals. II. A generalisation of the function,” Journal of Physics. A. Mathematical and General, vol. 20, no. 13, pp. 4119–4128, 1987.

R. G. Buschman and H. M. Srivastava, “The H function associated with a certain class of Feynman integrals,” Journal of Physics A: Mathematical and General, vol. 23, no. 20, pp. 4707–4710, 1990.

H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, India, 1982.

H. M. Srivastava, S. Gaboury, and A. Bayad, “Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus,” preprint 2014

S. Gaboury, “Some relations involving generalized Hurwitz-Lerch zeta function obtained by means of fractional derivatives with applications to Apostol-type polynomials,” Advances in Difference Equations, vol. 2013, article 361, pp. 1–13, 2013.

S. Gaboury and A. Bayad, “Series representations at special values of generalized Hurwitz-Lerch zeta function,” Abstract and Applied Analysis, vol. 2013, Article ID 975615, 8 pages, 2013.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.

A. Erdélyi, “An integral equation involving Legendre functions,” SIAM Journal on Applied Mathematics, vol. 12, pp. 15–30, 1964.

J. Liouville, “Mémoire sur le calcul des différentielles à indices quelconques,” Journal de l'École Polytechnique, vol. 13, pp. 71–162, 1832.

M. Riesz, “L'intégrale de Riemann-Liouville et le problème de Cauchy,” Acta Mathematica, vol. 81, pp. 1–223, 1949.

T. J. Osler, “The fractional derivative of a composite function,” SIAM Journal on Mathematical Analysis, vol. 1, pp. 288–293, 1970.

T. J. Osler, “Leibniz rule for fractional derivatives generalized and an application to infinite series,” SIAM Journal on Applied Mathematics, vol. 18, pp. 658–674, 1970.

T. J. Osler, Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives [Ph.D. thesis], New York University, 1970.

T. J. Osler, “Mathematical notes: fractional derivatives and Leibniz rule,” The American Mathematical Monthly, vol. 78, no. 6, pp. 645–649, 1971.

J.-L. Lavoie, R. Tremblay, and T. J. Osler, “Fundamental properties of fractional derivatives via Pochhammer integrals,” in Fractional Calculus and Its Applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974), B. Ross, Ed., vol. 457 of Lecture Notes in Math., pp. 323–356, Springer, Berlin, Germany, 1975.

R. Tremblay, Une Contribution à la Théorie de la Dérivée Fractionnaire [Ph.D. thesis], Laval University, Québec, Canada, 1974.

H. M. Srivastava, S.-D. Lin, and P.-Y. Wang, “Some fractional-calculus results for the -function associated with a class of Feynman integrals,” Russian Journal of Mathematical Physics, vol. 13, no. 1, pp. 94–100, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

R. Tremblay, S. Gaboury, and B.-J. Fugère, “A new Leibniz rule and its integral analogue for fractional derivatives,” Integral Transforms and Special Functions, vol. 24, no. 2, pp. 111–128, 2013.

T. J. Osler, “Taylor's series generalized for fractional derivatives and applications,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 37–48, 1971.

G. H. Hardy, “Riemann's form of Taylor's series,” Journal of the London Mathematical Society, vol. 20, pp. 48–57, 1945.

O. Heaviside, Electromagnetic Theory, vol. 2, Dover Publishing, New York, NY, USA, 1950.

B. Riemann, Versuch einer allgemeinen Auffasung der Integration und Differentiation, The Collected Works of Bernhard Riemann, Dover Publishing Company, New York, NY, USA, 1953.

Y. Watanabe, “Zum Riemanschen binomischen Lehrsatz,” Proceedings of the Physico-Mathematical Society of Japan, vol. 14, pp. 22–35, 1932.

R. Tremblay, S. Gaboury, and B.-J. Fugère, “Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives,” Integral Transforms and Special Functions, vol. 24, no. 1, pp. 50–64, 2013.

R. Tremblay and B.-J. Fugère, “The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 507–529, 2007.

S. Gaboury and R. Tremblay, “Summation formulas obtained by means of the generalized chain rule for fractional derivatives,” preprint 2014

H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1985.

H. M. Srivastava, D. Jankov, T. K. Pogány, and R. K. Saxena, “Two-sided inequalities for the extended Hurwitz-Lerch zeta function,” Computers & Mathematics with Applications, vol. 62, no. 1, pp. 516–522, 2011.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, McGraw-Hill Book, New York, NY, USA, 1953.

H. M. Srivastava, M.-J. Luo, and R. K. Raina, “New results involving a class of generalized Hurwitz-Lerch zeta functions and their applications,” Turkish Journal of Analysis and Number Theory, vol. 1, pp. 26–35, 2013.

H. M. Srivastava, “A new family of the -generalized Hurwitz-Lerch zeta functions with applications,” Applied Mathematics & Information Sciences, vol. 8, no. 4, pp. 1485–1500, 2014.