Abstract
We derive several new expansion formulas for a new family of the λ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems 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 considered.
1. Introduction
The Hurwitz-Lerch Zeta function which is one of the fundamentally important higher transcendental functions is defined by (see, e.g., [1, p. 121 et seq.]; see also [2] and [3, p. 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, but also such other important functions of Analytic Number Theory as the Polylogarithmic function (or de Jonquière’s function) : and the Lipschitz-Lerch zeta function (see [1, p. 122, Equation 2.5 (11)]): Indeed, the Hurwitz-Lerch zeta function defined in (5) can be continued meromorphically to the whole complex -plane, except for a simple pole at with its residue . It is also 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 (5) (see also [9]). In particular, they considered the following function: 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, p. 21 et seq.]). In terms of the extended Hurwitz-Lerch zeta function defined by (6), the following generalization of several known integral representations arising from (5) was given by Srivastava et al. [7] as follows: provided that the integral exists.
Definition 1. The function or involved in the right-hand side of (9) is the well-known Fox-Wright function, which is a generalization of the familiar generalized hypergeometric function , with numerator parameters and denominator parameters such that defined by (see, for details, [11, p. 21 et seq.] and [10, p. 50 et seq.]) where the equality in the convergence condition holds true for suitably bounded values of given by
Recently, Srivastava [12] introduced and investigated a significantly more general class of Hurwitz-Lerch zeta type functions by suitably modifying the integral representation formula (9). Srivastava considered the following function: so that, clearly, we have the following relationship:
In its special case when the definition (13) would reduce to the following form: where we have assumed further that or provided that the integral (16) exists. The function was studied by Raina and Chhajed [13, Eq. (1.6)] and, more recently, by Srivastava et al. [14].
As a particular interesting case of the function , we recall the following function: The function was introduced by Goyal and Laddha [4] as follows:
Another special case of the function that is worthy to mention occurs when and . We have where the function is the extended Hurwitz zeta function introduced by Chaudhry and Zubair [15].
In his work, Srivastava [12, p. 1489, Eq. (2.1)] also derived the following series representation of the function : provided that both sides of (22) exist.
Definition 2. The -function involved in the right-hand side of (22) is the well-known Fox’s -function defined by [16, Definition 1.1] (see also [10, 17]) where An empty product is interpreted as , , and are integers such that , , , , , , and is a suitable Mellin-Barnes type contour separating the poles of the gamma functions from the poles of the gamma functions
It is important to recall that Srivastava [12, p. 1490, Eq. (2.10)] presented another series representation for the function involving the Laguerre polynomials of order and degree in generated by (see, for details, [10]) Explicitly, it was proven by Srivastava [12, p. 1490, Eq. (2.10)] that provided that each member of (28) exists and being given by (9).
Motivated by a number of recent works by the present authors [18–20] and also of those of several other authors [4–9, 21, 22], this paper aims to provide many new relationships involving the new family of the -generalized Hurwitz-Lerch zeta function .
2. Pochhammer Contour Integral Representation for Fractional Derivative
The most familiar representation for the fractional derivative of order of is the Riemann-Liouville integral [23] (see also [24–26]); 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 [26]).
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, e.g., the works of Osler [27–30]).
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 Tremblay [31, 32].
Definition 3. 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 (32) at the beginning and the ending point of the Pochhammer contour which is closed on the Riemann surface of the multiple-valued function .
Remark 4. In Definition 3, 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 (32) would still be valid.
Remark 5. In case the Pochhammer contour never crosses the singularities at and in (32), 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 (32). However, by integrating by parts times the integral in (32) by two different ways, we can show that and are removable singularities (see, for details, [31]).
It is well known that [33, p. 83, Equation (2.4)] Adopting the Pochhammer based representation for the fractional derivative modifies the restriction to the case when is not a negative integer.
Now, by using (33) in conjunction with the series representation (22) for , we obtain the following important fractional derivative formula that will play an important role in our present investigation:
3. Important Results 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 a fundamental formula related to the generalized chain rule for fractional derivatives, the Taylor-like expansions in terms of different types of functions, and the generalized Leibniz rules for fractional derivatives.
First of all, Osler [27, p. 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 6 below.
Theorem 6. 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:
Relation (35) allows us to obtain very easily known and new summation formulas involving special functions of mathematical physics.
By applying the relation (35), Gaboury and Tremblay [34] proved the following corollary which will be useful in the next section.
Corollary 7. Under the hypotheses of Theorem 6, let be a positive integer. Then the following relation holds true: where
Next, in the year 1971, Osler [35] obtained the following generalized Taylor-like series expansion involving fractional derivatives.
Theorem 8. Let be an analytic function in a simply-connected region . Let and be arbitrary complex numbers and let with a regular and univalent function without any zero in . Let be a positive real number and let Let and be two points in such that and let Then the following relationship holds true:
In particular, if and , then and the formula (41) reduces to the following form: This last formula (42) is usually referred to as the Taylor-Riemann formula and has been studied in several papers [29, 36–39].
We next recall that Tremblay et al. [40] discovered 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 9. (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 (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 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 3 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 9 reduces to the following form:
Tremblay and Fugère [41] 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 gave the following theorem.
Theorem 10. Under the assumptions of Theorem 9, the following expansion formula holds true where
As a special case, if we set , , and in (55), we obtain
Finally, we give two generalized Leibniz rules for fractional derivatives. Theorem 11 is a slightly modified theorem obtained in 1970 by Osler [28]. Theorem 12 was given, some years ago, by Tremblay et al. [42] with the help of the properties of Pochhammer’s contour representation for fractional derivatives.
Theorem 11. (i) Let be a simply-connected region containing the origin. (ii) Let and satisfy the conditions of Definition 3 for the existence of the fractional derivative. Then, for and , the following Leibniz rule holds true
Theorem 12. (i) Let be a simply-connected region containing the origin. (ii) Let and satisfy the conditions of Definition 3 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:
4. Main Expansion Formulas
This section is devoted to the presentation of the new relations involving the new family of the -generalized Hurwitz-Lerch zeta function .
Theorem 13. Under the hypotheses of Corollary 7, let be a positive integer. Then the following relation holds true: where and denotes the Lauricella function of variables defined by [11, p. 60] provided that both sides of (61) exist.
Proof. Putting and letting in Corollary 7, we get
With the help of the definition of given by (37), we find for the left-hand side of (63) that
We now expand each factor in the product in (63) in power series and replace the generalized Hurwitz-Lerch zeta function by its -function series representation. We, thus, find for the right-hand side of (63) that
Finally, by combining (64) and (65), we obtain the result (61) asserted by Theorem 13.
We now shift our focus on the different Taylor-like expansions in terms of different types of functions involving the new family of the -generalized Hurwitz-Lerch zeta function .
Theorem 14. Under the assumptions of Theorem 8, the following expansion formula holds true provided that both members of (66) exist.
Proof. Setting in Theorem 8 with , , and , we have
for and for such that .
Now, by making use of (34) with and , we find that
By combining (67) and (68), we get the result (66) asserted by Theorem 14.
Theorem 15. Under the hypotheses of Theorem 9, the following expansion formula holds true: for and for on defined by provided that both sides of (69) exist.
Proof. By taking in Theorem 9 with , , , and , we find that Now, with the help of the relation (34) with and , we have Thus, by combining (71) and (72), we are led to the assertion (69) of Theorem 15.
Theorem 16. Under the hypotheses of Theorem 10, the following expansion formula holds true: for and for on defined by provided that both sides of (73) exist.
Proof. Putting in Theorem 10 with , , , and , we find that With the help of the relations in (34), we have Thus, by combining (75) and (76), we obtain the desired result (73).
Finally, from the two generalized Leibniz rules for fractional derivatives given in Section 3, we obtain the following two expansion formulas involving the new family of the -generalized Hurwitz-Lerch zeta function .
Theorem 17. Under the hypotheses of Theorem 11, the following expansion formula holds true provided that both members of (77) exist.
Proof. Setting and in Theorem 11 with and , we obtain which, with the help of (33) and (34), yields Combining (79) with (78) and making some elementary simplifications, the asserted result (77) follows.
Theorem 18. Under the hypotheses of Theorem 12, the following expansion formula holds true provided that both members of (80) exist.
Proof. Upon first substituting and in Theorem 12 and then setting in which both and satisfy the conditions of Theorem 12, we have Now, by using (33) and (34), we find that Thus, finally, the result (80) follows by combining (83) and (82).
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.
Setting in Theorem 13 and using the fact that [12, p. 1496, Remark 7] we obtain the following corollary given recently by Srivastava et al. [19].
Corollary 19. Under the hypotheses of Theorem 13, the following expansion formula holds true where denotes the first Appell function defined by [11, p. 22] provided that both sides of (85) exist.
Putting and setting and in Theorem 15 reduces to the following expansion formula given recently by Srivastava et al. [20].
Corollary 20. Under the hypotheses of Theorem 15, the following expansion formula holds true: for and for on defined by provided that both sides of (69) exist.
Letting in Theorem 18, we deduce the following expansion formula obtained by Srivastava et al. [19].
Corollary 21. Under the hypotheses of Theorem 18, the following expansion formula holds true provided that both members of (89) exist.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.