We begin this section by recalling the following sum-integral representation given by Yen et al.
), p. 100, Theorem) for the Hurwitz (or generalized) Zeta function ζ
) defined by (1.3):
which, for k
= 2, was derived earlier by Nishimoto et al.
), p. 94, Theorem 4). The following straightforward generalization of the sum-integral representation (2.1) involving the familiar general Hurwitz-Lerch Zeta function Φ(z
) defined by (1.1) was given by Lin and Srivastava (2004
, p. 727, Equation (7
The sum-integral representations (2.1) and (1.2) led Lin and Srivastava (2004
) to the introduction and investigation of an interesting generalization of the Hurwitz-Lerch Zeta function Φ(z
) in the following form given by Lin and Srivastava ((2004
), p. 727, Equation (8
denotes the Pochhammer symbol defined, in terms of the familiar Gamma function, by (1.8). Clearly, we find from the definition (2.3) that
where, as already pointed out by Lin and Srivastava (2004), is a generalization of the Hurwitz-Lerch Zeta function considered by Goyal and Laddha ((1997), p. 100, Equation (1.5)). For further results involving these classes of generalized Hurwitz-Lerch Zeta functions, see the recent works by Garg et al. (2006) and Lin et al.(2006).
A generalization of the above-defined Hurwitz-Lerch Zeta functions Φ(z
was studied, in the following form, by Garg et al.
), p. 313, Equation (1
Various integral representations and two-sided bounding inequalities for Φλ, μ;ν(z, s, a) can be found in the works by Garg et al. (2008) and [Jankov et al.(2011)], respectively. These latter authors [Jankov et al.(2011)] also considered the function Φλ, 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 Φλ, μ;ν
) studied by Garg et al.
) does not
provide a generalization of the function
which was introduced earlier by Lin and Srivastava (2004
). Indeed, for λ
= 1, the function Φλ, μ;ν
) coincides with a special
case of the function
= 1, that is,
Next, for the Riemann-Liouville fractional derivative operator
defined by (see, for example, Erdélyi et al.
), p. 181), Samko et al.
) and (Kilbas et al. 2006
, p. 70 et seq.
the following formula is well-known:
which, by virtue of the definitions (1.1) and (2.3), yields the following fractional derivative formula for the generalized Hurwitz-Lerch Zeta function with ρ
[Lin and Srivastava ((2004
), p. 730, Equation (24
In its particular case when ν
= 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
is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz-Lerch function Φ(z
). Moreover, it is easily deduced from the fractional derivative formula (2.8) that
which (as observed recently by Srivastava et al. (2011), pp. 490–491) exhibits the fact that the function Φλ, μ;ν(z, s, a) studied by Garg et al. (2008) is essentially a consequence of the classical Hurwitz-Lerch Zeta function Φ(z, s, a) when we apply the Riemann-Liouville fractional derivative operator two times as indicated above in (2.11). The interested reader may be referred also to many other explicit representations for and , 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, n-dimentional) Hurwitz-Lerch Zeta function Φ
(z, s, a) was studied recently by Choi et al. ((2008), p. 66, Eq. (6)). On the other hand, Răducanu and Srivastava (see (Răducanu and Srivastava 2007), the references cited therein as well as many sequels thereto) made use of the Hurwitz-Lerch Zeta function Φ(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 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 Srivastava et al. (2011)), fruitfully using the Hurwitz-Lerch Zeta function Φ(z, s, a) and some of its above-mentioned generalizations, have led eventually to the following definition a family of the extended (multi-parameter) Hurwitz-Lerch Zeta functions by Srivastava et al. (2011).
(Srivastava et al.
)). The family of the extended (multi-parameter) Hurwitz-Lerch Zeta functions
is defined by
where the sequence
of the coefficients in the definition (2.12) is given, for latter convenience, by
denotes the Pochhammer symbol given by (1.8) and
In order to derive direct
relationships of the family of the extended (multi-parameter) Hurwitz-Lerch Zeta functions
defined by (2.12) with several other relatively more familiar special functions, we need each of the following definitions.
The Fox-Wright function
, which is a further
generalization of the familiar generalized hypergeometric function
, with p
numerator parameters a1
, ⋯, a
denominator parameters b1
, ⋯, b
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
In the particular case when
we have the following relationship (see, for details, (Choi et al.1985
, p. 21)):
in terms of the generalized hypergeometric function .
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
can take on noninteger values, and
is a Mellin-Barnes type contour starting at the point τ
and terminating at the point τ
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 Λ is defined by (2.19).