### Abstract

Motivated essentially by recent works by several authors (see, for example, Bin-Saad
[Math J Okayama Univ 49:37–52, 2007] and Katsurada [Publ Inst Math (Beograd) (Nouvelle
Ser) 62(76):13–25, 1997], the main objective in this paper is to present a systematic
investigation of numerous interesting properties of some families of generating functions
and their partial sums which are associated with various classes of the extended Hurwitz-Lerch
Zeta functions. Our main results would generalize and extend the aforementioned recent
work by Bin-Saad [Math J Okayama Univ 49:37–52, 2007] (see also Katsurada [Publ Inst
Math (Beograd) (Nouvelle Ser) 62(76):13–25, 1997]). 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 [Jñānābha 41:63–68, 2011]) and Saxena *et al.* [J Indian Acad Math 33:309–320, 2011], is simply a seemingly trivial notational variation
of the familiar general Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*). Finally, we present a sum-integral representation formula for the general family
of the extended Hurwitz-Lerch Zeta functions.

#### 2010 Mathematics Subject Classification

Primary 11M25, 33C60; Secondary 33C05

##### Keywords:

Riemann; Hurwitz (or generalized) and Hurwitz-Lerch Zeta functions; Lerch Zeta function and the Polylogarithmic (or de Jonquière’s) function; General Hurwitz-Lerch Zeta function; Gauss and Kummer hypergeometric functions; Fox-Wright*Ψ*-function and the

### Introduction and preliminaries

Throughout our present investigation, we use the following standard notations:

and

Here, as usual,
*positive* real numbers and

The familiar general Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*) defined by (see, for example, (Erdélyi *et al.*1953, p. 27. Eq. 1.11 (1)); see also Srivastava and Choi ((2001, p. 121 *et seq.*) and (Srivastava and Choi 2012), p. 194 *et seq.*)

contains, as its *special* cases, not only the Riemann Zeta function *ζ*(*s*), the Hurwitz (or generalized) Zeta function *ζ*(*s*,*a*) and the Lerch Zeta function *ℓ*_{s}(*ξ*) defined by (see, for details, (Erdélyi *et al.*1953, Chapter I) and Srivastava and Choi ((2001), Chapter 2)

and

respectively, but also such other important functions of *Analytic Number Theory* as the Polylogarithmic function (or *de Jonquière’s function*) Li_{s}(*z*):

and the Lipschitz-Lerch Zeta function *ϕ*(*ξ*,*a*,*s*) (see Srivastava and Choi ((2001), p. 122, Equation 2.5 (11))):

which was first studied by Rudolf Lipschitz (1832-1903) and Matyáš Lerch (1860-1922)
in connection with Dirichlet’s famous theorem on primes in arithmetic progressions
(see also (Srivastava 2011), Section 5). Indeed, just as its aforementioned special cases *ζ*(*s*) and *ζ*(*s*,*a*), the Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*) defined by (1.1) can be continued *meromorphically* to the whole complex *s*-plane, except for a simple pole at *s* = 1 with its residue 1. It is also known that (Erdélyi *et al.*1953, p. 27, Equation 1.11 (3))

Making use of the Pochhammer symbol (or the *shifted* factorial)

it being understood *conventionally* that (0)_{0}: = 1 and assumed *tacitly* that the Gamma quotient exists, we recall each of the following well-known expansion
formulas:

and

More generally, it is not difficult to show similarly that

which would reduce immediately to the expansion formula (1.10) in its special case
when *λ* = *s*. Moreover, in the limit case when

this last result (1.11) yields

Wilton (1922/1923) applied the expansion formula (1.9) in order to rederive Burnside’s formula (Erdélyi
*et al.*1953, p. 48, Equation 1.18 (11)) for the sum of a series involving the Hurwitz (or generalized)
Zeta function *ζ*(*s*,*a*). Srivastava (see, for details, Srivastava (1988a;1988b)), on the other hand, made use of such expansion formulas as (1.9) and (1.10) as
well as the obvious special case of (1.9) when *a*=1 for finding the sums of various classes of series involving the Riemann Zeta function
*ζ*(*s*) and the Hurwitz (or generalized) Zeta function *ζ*(*s*,*a*) (see also Srivastava and Choi ((2001), Chapter 3) and (Srivastava and Choi 2012), Chapter 3).

Various results for the generating functions *ϑ*_{λ}(*z*,*t*;*s*,*a*) and *φ*(*z*,*t*;*s*,*a*), which are defined by (1.11) and (1.12), respectively, were given recently by Bin-Saad
(2007, p. 46, Equations (5.1) to (5.4)) who also considered each of the following *truncated* forms of these generating functions:

and

so that, obviously, we have

and

For the Riemann Zeta function *ζ*(*s*), the special case of each of the generating functions *ϑ*_{λ}(*z*,*t*;*s*,*a*) and *φ*(*z*,*t*;*s*,*a*) in (1.11) and (1.12) when *z* = *a* = 1 was investigated by Katsurada (1997). Subsequently, various results involving the generating functions *ϑ*_{λ}(*z*,*t*;*s*,*a*) and *φ*(*z*,*t*;*s*,*a*) defined by (1.11) and (1.12), respectively, together with their such partial sums
as those given by (1.13) to (1.16), were derived by Bin-Saad (2007) (see also the more recent sequels to (Bin-Saad 2007) and (Katsurada 1997) by Gupta and Kumari (2011) *and* by Saxena *et al.* (2011a).

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 Φ(*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.

### Families of the extended Hurwitz-Lerch Zeta functions and related special functions

We begin this section by recalling the following sum-integral representation given
by Yen *et al.* ((2002), p. 100, Theorem) for the Hurwitz (or generalized) Zeta function *ζ*(*s*,*a*) defined by (1.3):

which, for *k* = 2, was derived earlier by Nishimoto *et al.*((2002), 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*,*s*,*a*) 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*,*s*,*a*) in the following form given by Lin and Srivastava ((2004), p. 727, Equation (8)):

where (*λ*)_{ν} denotes the Pochhammer symbol defined, in terms of the familiar Gamma function, by
(1.8). Clearly, we find from the definition (2.3) that

and

where, as already pointed out by Lin and Srivastava (2004),
*et al.* (2006) and Lin *et al.*(2006).

A generalization of the above-defined Hurwitz-Lerch Zeta functions Φ(*z*,*s*,*a*) and
*et al.*((2008), p. 313, Equation (1.7)):

Various integral representations and two-sided bounding inequalities for Φ_{λ,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 Φ_{λ,μ;ν}(*z*,*s*,*a*) as a special kind of Mathieu type (** a**,

**)-series.**

*λ*If we compare the definitions (2.3) and (2.6), we can easily observe that the function
Φ_{λ,μ;ν}(*z*,*s*,*a*) studied by Garg *et al.*(2008) does *not* provide a generalization of the function
*λ* = 1, the function Φ_{λ,μ;ν}(*z*,*s*,*a*) coincides with a *special* case of the function
*ρ* = *σ* = 1, that is,

Next, for the *Riemann-Liouville fractional derivative operator*
*et al.*((1954), p. 181), Samko *et al.*(1993) 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* = *σ* [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
*z*,*s*,*a*). 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 Φ_{λ,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

It should be remarked here that a multiple (or, simply, *n*-dimentional) Hurwitz-Lerch Zeta function Φ_{n}(*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).

#### Definition 1

(Srivastava *et al.* (2011)). The family of the extended (multi-parameter) Hurwitz-Lerch Zeta functions

is defined by

where the sequence

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.

#### Definition 2

The Fox-Wright function
*further* generalization of the familiar generalized hypergeometric function
*p* numerator parameters *a*_{1}, ⋯,*a*_{p} and *q* denominator parameters *b*_{1}, ⋯,*b*_{q} such that

and

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

#### 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

the exponents

can take on noninteger values, and
*τ* − i*∞* and terminating at the point *τ* + i*∞*

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).

### Remark 1

If we set

and

then (2.12) reduces to the following generalized *M*-series which was recently introduced by Sharma and Jain (2009) (see also an earlier paper by Sharma (2008) for the *special* case when *β*=1):

in which the last relationship exhibits the fact that the so-called generalized *M*-series is indeed an *obvious* special case of the Fox-Wright function
*et al.* (2002), we have

### Remark 2

The following

from those of the
*et al.* ((2011), p. 504, Theorem 8)):

the path of integration
*ξ*-plane, which starts at the point −i*∞* and terminates at the point i*∞* with indentations, if necessary, in such a manner as to separate the poles of *Γ*(−*ξ*) from the poles of *Γ*(*λ*_{j}+*ρ*_{j}*ξ*) (*j*=1,⋯,*p*). Thus, for example, by making use of a known fractional-calculus result due to Srivastava
*et al.* ((2006), p. 97, Equation (2.4)), we readily obtain the following extension of such fractional
derivative formulas as (2.9) and (2.10) [Srivastava *et al.* ((2011), p. 505, Equation (6.8))]:

### Generating relations associated with the extended Hurwitz-Lerch Zeta function

In this section, we first introduce the following generating functions and their partial sums involving the extended Hurwitz-Lerch Zeta function

defined by (2.12). Indeed, as a generalization of the generating functions (1.9) and (1.10), we have

which can easily be put in the following more general form:

where the sequence
*λ* = *s*. Furthermore, in its limit case when

the generating function (3.2) yields

where the sequence

We shall also consider each of the following *truncated* forms of the generating functions *Ω*_{λ}(*z*,*t*;*s*,*a*) and *Θ*(*z*,*t*;*s*,*a*) in (3.2) and (3.3), respectively:

and

which obviously satisfy the following decomposition formulas:

and

Our first set of integral representations for the above-defined generating functions is contained in Theorem 1 below.

#### Theorem 1

Each of the following integral representation formulas holds true:

and

provided that both sides of each of the assertions (3.10) and (3.11) exist.

#### Proof

For convenience, we denote by
_{1}*F*_{1} in series forms, we find that

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), Ξ_{n} being defined by (2.13). Now, if we evaluate the innermost integral in (3.12) by
appealing to the following well-known result:

we get

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. □

#### Remark 3

For *ω* = 0, each of the assertions (3.10) and (3.11) of Theorem 1 yields a known integral
representation formula due to Srivastava *et al.* ((2011), p. 504, Equation (6.4)). Moreover, in their special case when

the assertions (3.10) and (3.11) of Theorem 1 would reduce immediately to the classical
integral representation (1.7) for the Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*).

The proof of Theorem 2 below would run parallel to that of Theorem 1, which we already
have detailed above fairly adequately. It is based essentially upon the Hankel type
contour integral in the following form (Erdélyi *et al.*1953, p. 14, Equation 1.16 (4)):

or, equivalently,

#### Theorem 2

Each of the following Hankel type contour integral representation formulas holds true :

and

provided that both sides of each of the assertions (3.17) and (3.18) exist.

#### Remark 4

For *ω* = 0, each of the assertions (3.17) and (3.18) of Theorem 2 yields the following (*presumably new*) integral representation formula:

Furthermore, in their special case when

the assertions (3.17) and (3.18) of Theorem 2 would reduce to the classical Hankel
type contour integral representation for the Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*) (see, for example, (, Erdélyi *et al.*1953, p. 28, Equation 1.11 (5)); see also (Srivastava and Choi 2012), p. 195, Equation 2.5 (8)).

Next, by making use of the following known result (see, for example, Srivastava and Manocha ((1984), p. 86, Problem 1):

we evaluate several Eulerian Beta-function integrals involving the generating functions
*Ω*_{λ}(*z*,*t*;*s*,*a*) and *Θ*(*z*,*t*;*s*,*a*) defined by (3.2) and (3.3), respectively, *B*(*α*,*β*) being the familiar Beta function.

#### Theorem 3

In terms of the sequence

and

provided that both sides of each of the assertions (3.21) and (3.22) exist, the Fox-Wright
function 00
*H*-function contained in the definition (2.18).

#### 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 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 *Ω*_{λ}(*z*,*t*;*s*,*a*) and *Θ*(*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 well-known formula (3.13).

#### Theorem 4

Let the function

and

provided that both sides of each of the assertions (3.23), (3.24) and (3.25) exist.

#### Remark 6

Some very specialized cases of Theorem 4 when

were derived in the recent works (Bin-Saad 2007), (Gupta and Kumari 2011) and (Saxena *et al.* (2011a)).

#### 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 left-hand sides obviously
violate the required convergence conditions at their *lower* terminal *t* = 0.

We now turn toward the *truncated* forms of the generating functions *Ω*_{λ}(*z*,*t*;*s*,*a*) and *Θ*(*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

and

provided that both sides of each of the assertions (3.26) to (3.29) exist.

#### Remark 8

Several specialized cases of Theorem 5 when

can be found in the recent works (Bin-Saad 2007), (Gupta and Kumari 2011) and ( (Saxena *et al.*2011a)).

It is not difficult to derive various other properties and results involving the generating
functions *Ω*_{λ}(*z*,*t*;*s*,*a*) and *Θ*(*z*,*t*;*s*,*a*) in (3.2) and (3.3), respectively, as well as their *truncated* forms which are defined by (3.4) to (3.7). For example, by applying the definition
(3.2) in conjunction with the definition (2.15), it is easy to derive the following
general form of the generating relations asserted by (for example) Bin-Saad ( 2007, p. 44, Theorem 4.2):

where the sequence
*tacitly* assumed that each member of the generating relation (3.30) exists. We do, however,
choose to leave the details involved in all such derivations as exercises for the
interested reader.

*τ*-Generalizations of the Hurwitz-Lerch Zeta functions

In a recent paper, Saxena *et al.* (2011a) considered a so-called *τ*-generalization of the Hurwitz-Lerch Zeta function Φ(*z*,*s*,*a*) in (1.1) in the following form [Saxena *et al.* ((2011a), p. 311, Equation (2.1))]:

Subsequently, by similarly introducing a parameter *τ*>0 in the definition (2.5), Gupta and Kumari (2011) studied a *τ*-generalization of the extended Hurwitz-Lerch Zeta function

which, when compared with the definition (4.1), yields the relationship:

By looking closely at the definitions (4.1) and (4.2), in conjunction with the earlier definitions (1.1) and (2.5), respectively, we immediately get the following rather obvious connections:

and

Clearly, therefore, the definitions in (4.1) and (4.2) (*with*
*τ* = 1 given by the definitions in (1.1) and (2.5), respectively. Thus, by *trivially* appealing to the parametric changes exhibited by the connections in (4.4) and (4.5),
all of the results involving the so-called *τ*-generalized functions Φ(*τ*;*z*,*s*,*a*) and
*z*,*s*,*a*) and
*et al.* ((2011), p. 494, Equation (2.6)) for the special case when *k* = 1):

it being tacitly assumed that each member of (4.6) exists. Indeed, in the special
case when *ρ*=*σ*=*ν*=1, (4.6) yields the following sum-integral representation for the generalized Hurwitz-Lerch
Zeta function

or, equivalently,

where, for convenience, *Δ*^{∗}(*n*;*λ*) abbreviates the array of *n* parameters

the array being empty when *n* = 0.

Now, in order to rewrite this last result (4.8) in terms of the *τ*-generalized Hurwitz-Lerch Zeta function

and multiply the resulting equation by *τ*^{−s}. By using the connection in (4.5), we thus find immediately that