Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions

2010 Mathematics Subject Classification Primary 11M25, 33C60; Secondary 33C05

Our main objective in this paper is to investigate, in a rather systematic manner, much more general families of generating functions and their partial sums than those associated with the generating functions ϑ λ (z, t; s, a) and ϕ(z, t; s, a) defined by (1.11) and (1.12), respectively. We also show the hitherto unnoticed fact that the so-called τ -generalized Riemann Zeta function, which happens to be the main subject of investigation by Gupta and Kumari (2011) and by Saxena et al. (2011a), is simply a seemingly trivial notational variation of the familiar general Hurwitz-Lerch Zeta function (z, s, a) defined by (1.1). Finally, we present a sum-integral representation formula for the general family of the extended Hurwitz-Lerch Zeta functions.
A generalization of the above-defined Hurwitz-Lerch Zeta functions (z, s, a) and * μ (z, s, a) was studied, in the following form, by Garg et al. ((2008), p. 313, Equation (1.7)): Various integral representations and two-sided bounding inequalities for λ,μ;ν (z, s, a) can be found in the works by Garg et al. (2008) and ], respectively. These latter authors ] also considered the function λ,μ;ν (z, s, a) as a special kind of Mathieu type (a, λ)-series.
If we compare the definitions (2.3) and (2.6), we can easily observe that the function λ,μ;ν (z, s, a) studied by Garg et al. (2008) does not provide a generalization of the function (ρ,σ ) μ,ν (z, s, a) which was introduced earlier by Lin and Srivastava (2004). Indeed, for λ = 1, the function λ,μ;ν (z, s, a) coincides with a special case of the function Next, for the Riemann-Liouville fractional derivative operator D μ z defined by (see, for example, Erdélyi et al. ((1954), p. 181), Samko et al. (1993) and (Kilbas et al. 2006, p. 70 the following formula is well-known:  [Lin and Srivastava ((2004), p. 730, Equation (24))]: In its particular case when ν = σ = 1, the fractional derivative formula (2.9) would reduce at once to the following form: * which (as already remarked by Lin and Srivastava (2004), p. 730) exhibits the interesting (and useful) fact that * μ (z, s, a) is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz-Lerch function (z, s, a). Moreover, it is easily deduced from the fractional derivative formula (2.8) that which (as observed recently by , pp. 490-491) exhibits the fact that the function λ,μ;ν (z, s, a) studied by Garg et al. (2008) is essentially a consequence of the classical Hurwitz-Lerch Zeta function (z, s, a) when we apply the Riemann-Liouville fractional derivative operator D μ z two times as indicated above in (2.11). The interested reader may be referred also to many other explicit representations for * μ (z, s, a) and (ρ,σ ) μ,ν (z, s, a), which were proven by Lin and Srivastava (2004), including (for example) a potentially useful Eulerian integral representation of the first kind [Lin and Srivastava ((2004), p. 731, Equation (28))].
It should be remarked here that a multiple (or, simply, ndimentional) Hurwitz-Lerch Zeta function n (z, s, a) was studied recently by Choi et al. ((2008), p. 66, Eq. (6)). On the other hand, Rȃducanu and Srivastava (see (Rȃducanu and Srivastava 2007), the references cited therein as well as many sequels thereto) made use of the Hurwitz-Lerch Zeta function (z, s, a) in defining a certain linear convolution operator in their systematic investigation of various analytic function classes in Geometric Function Theory in Complex Analysis. Furthermore, Gupta http://www.springerplus.com/content/2/1/67 et al. (2008) revisited the study of the familiar Hurwitz-Lerch Zeta distribution by investigating its structural properties, reliability properties and statistical inference. These investigations by Gupta et al. (2008) and others (see, for example, (Srivastava 2000), Srivastava and Choi (2001) and Srivastava et al. (2010); see also Saxena et al. (2011b) and ), fruitfully using the Hurwitz-Lerch Zeta function (z, s, a) and some of its above-mentioned generalizations, have led eventually to the following definition a family of the extended (multiparameter) Hurwitz-Lerch Zeta functions by .
, which is a further generalization of the familiar generalized hypergeometric function p F q (p, q ∈ N 0 ), with p numerator parameters a 1 , · · · , a p and q denominator parameters b 1 , · · · , b q such that defined by (see, for details, (Erdélyi et al. 1953, p. 183) and (Choi et al. 1985, p. 21); see also (Kilbas et al. 2006, p. 56), (Choi et al. 2010, p. 30) and (Srivastava et al. 1982, p. 19 where the equality in the convergence condition holds true for suitably bounded values of |z| given by (2.16) In the particular case when A j = B k = 1 (j = 1, · · · , p; k = 1, · · · , q), http://www.springerplus.com/content/2/1/67 we have the following relationship (see, for details, (Choi et al. 1985, p. 21)): p * q ⎡ ⎣ (a 1 , 1) , · · · , a p , 1 ; 1) , · · · , a p , 1 ; (2.17) in terms of the generalized hypergeometric function Definition 3. An attempt to derive Feynman integrals in two different ways, which arise in perturbation calculations of the equilibrium properties of a magnetic mode of phase transitions, led naturally to the following generalization of Fox's H-function (Inayat-Hussain 1987b, p. 4126) (see also (Buschman and Srivastava 1990) and (Inayat-Hussain 1987a)): which contains fractional powers of some of the Gamma functions involved. Here, and in what follows, the parameters A j > 0 (j = 1, · · · , p) and B j > 0 (j = 1, · · · , q), the exponents α j (j = 1, · · · , n) and β j (j = m + 1, · · · , q) can take on noninteger values, and L = L (iτ ;∞) is a Mellin-Barnes type contour starting at the point τ − i∞ and terminating at the point τ + i∞ (τ ∈ R) with the usual indentations to separate one set of poles from the other set of poles. The sufficient condition for the absolute convergence of the contour integral in (2.18) was established as follows by Buschman and Srivastava ((1990), p. 4708): which provides exponential decay of the integrand in (2.18) and the region of absolute convergence of the contour integral in (2.18) is given by where is defined by (2.19).
(2.24) http://www.springerplus.com/content/2/1/67 where the sequence { n } n∈N 0 of the coefficients in (2.12) is given, as before, by (2.13). We shall also consider each of the following truncated forms of the generating functions λ (z, t; s, a) and (z, t; s, a) in (3.2) and (3.3), respectively: which obviously satisfy the following decomposition formulas: Our first set of integral representations for the abovedefined generating functions is contained in Theorem 1 below.

11)
provided that both sides of each of the assertions (3.10) and (3.11) exist.
Proof. For convenience, we denote by S the second member of the assertion (3.10) of Theorem 1. Then, upon expanding the functions p * q and 1 F 1 in series forms, we find that (μ 1 , σ 1 ), · · · , (μ q , σ q ); where the inversion of the order of integration and double summation can easily be justified by absolute convergence under the conditions stated with (3.10), n being defined by (2.13). Now, if we evaluate the innermost integral in (3.12) by appealing to the following well-known result: which, in light of the definitions (2.12) and (3.2), yields the left-hand side of the first assertion (3.10) of Theorem 1.
The second assertion (3.11) of Theorem 1 can be proven in a similar manner.
Theorem 3. In terms of the sequence { n } n∈N 0 of the coefficients given by the definition (2.13), each of the following Eulerian Beta-function integral formulas holds true: Proof. Each of the assertions (3.21) and (3.22) of Theorem 3 can be proven fairly easily by appealing to the definitions (3.2) and (3.3), respectively, in conjunction http://www.springerplus.com/content/2/1/67 with the Eulerian Beta-function integral (3.20). The details involved are being skipped here.
Remark 5. In addition to their relatively more familiar cases when ξ = η − 1 = 0, various interesting limit cases of the integral formulas (3.21) and (3.22) asserted by Theorem 3 can be deduced by letting Some such very specialized cases of Theorem 3 can be found in the recent works by Bin-Saad (2007), Gupta and Kumari (2011) and Saxena et al. (2011a).
The Eulerian Gamma-function integrals involving the generating functions λ (z, t; s, a) and (z, t; s, a) defined by (3.2) and (3.3), respectively, which are asserted by Theorem 4 below, can be evaluated by applying the wellknown formula (3.13).
Theorem 4. Let the function * μ (z, s, a) be defined by (2.5). Then, in terms of the sequence { n } n∈N 0 of the coefficients given by the definition (2.13), each of the following single or double Eulerian Gamma-function integral formulas holds true: and provided that both sides of each of the assertions (3.23), (3.24) and (3.25) exist.
Remark 7. Two of the claimed integral formulas in Bin-Saad's paper (2007, p. 42, Theorem 3.2, Equations (3.10) and (3.11)) can easily be shown to be divergent, simply because the improper integrals occurring on their lefthand sides obviously violate the required convergence conditions at their lower terminal t = 0.
We now turn toward the truncated forms of the generating functions λ (z, t; s, a) and (z, t; s, a) in (3.2) and (3.3), respectively, which are defined by (3.4) to (3.7). Indeed, by appealing appropriately to the definitions in (3.4) to (3.7) in conjunction with the Eulerian Gamma-function integral in (3.13), it is fairly straightforward to derive the integral representation formulas asserted by Theorem 5 below.
Theorem 5. In terms of the sequence { n } n∈N 0 of the coefficients given by the definition (2.13), each of the following Eulerian Gamma-function integral formulas holds true: can be found in the recent works (Bin-Saad 2007), (Gupta and Kumari 2011) and (Saxena et al. (2011a)).