Abstract
Motivated essentially by recent works by several authors (see, for example, BinSaad [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 HurwitzLerch Zeta functions. Our main results would generalize and extend the aforementioned recent work by BinSaad [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 socalled τgeneralized 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 HurwitzLerch Zeta function Φ(z,s,a). Finally, we present a sumintegral representation formula for the general family of the extended HurwitzLerch Zeta functions.
2010 Mathematics Subject Classification
Primary 11M25, 33C60; Secondary 33C05
Keywords:
Riemann; Hurwitz (or generalized) and HurwitzLerch Zeta functions; Lerch Zeta function and the Polylogarithmic (or de Jonquière’s) function; General HurwitzLerch Zeta function; Gauss and Kummer hypergeometric functions; FoxWright Ψfunction and the function; MittagLeffler type functions; MellinBarnes type integral representations and Meromorphic continuation; Generating functions and Eulerian Gammafunction and Betafunction integral representationsIntroduction and preliminaries
Throughout our present investigation, we use the following standard notations:
and
Here, as usual, denotes the set of integers, denotes the set of real numbers, denotes the set of positive real numbers and denotes the set of complex numbers.
The familiar general HurwitzLerch 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 LipschitzLerch Zeta function ϕ(ξ,a,s) (see Srivastava and Choi ((2001), p. 122, Equation 2.5 (11))):
which was first studied by Rudolf Lipschitz (18321903) and Matyáš Lerch (18601922) 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 HurwitzLerch Zeta function Φ(z,s,a) defined by (1.1) can be continued meromorphically to the whole complex splane, 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) defined, in terms of the familiar Gamma function, by
it being understood conventionally that (0)_{0}: = 1 and assumed tacitly that the Gamma quotient exists, we recall each of the following wellknown 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 BinSaad (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 BinSaad (2007) (see also the more recent sequels to (BinSaad 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 socalled τgeneralized 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 HurwitzLerch Zeta function Φ(z,s,a) defined by (1.1). Finally, we present a sumintegral representation formula for the general family of the extended HurwitzLerch Zeta functions.
Families of the extended HurwitzLerch Zeta functions and related special functions
We begin this section by recalling the following sumintegral 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 sumintegral representation (2.1) involving the familiar general HurwitzLerch Zeta function Φ(z,s,a) defined by (1.1) was given by Lin and Srivastava (2004, p. 727, Equation (7)):
The sumintegral representations (2.1) and (1.2) led Lin and Srivastava (2004) to the introduction and investigation of an interesting generalization of the HurwitzLerch 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), is a generalization of the HurwitzLerch Zeta function considered by Goyal and Laddha ((1997), p. 100, Equation (1.5)). For further results involving these classes of generalized HurwitzLerch Zeta functions, see the recent works by Garg et al. (2006) and Lin et al.(2006).
A generalization of the abovedefined HurwitzLerch Zeta functions Φ(z,s,a) and was studied, in the following form, by Garg et al.((2008), p. 313, Equation (1.7)):
Various integral representations and twosided 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,λ)series.
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 which was introduced earlier by Lin and Srivastava (2004). Indeed, for λ = 1, the function Φ_{λ,μ;ν}(z,s,a) coincides with a special case of the function when ρ = σ = 1, that is,
Next, for the RiemannLiouville fractional derivative operator defined by (see, for example, Erdélyi et al.((1954), p. 181), Samko et al.(1993) and (Kilbas et al.2006, p. 70 et seq.))
the following formula is wellknown:
which, by virtue of the definitions (1.1) and (2.3), yields the following fractional derivative formula for the generalized HurwitzLerch Zeta function withρ = 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 is essentially a RiemannLiouville fractional derivative of the classical HurwitzLerch function Φ(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 HurwitzLerch Zeta function Φ(z,s,a) when we apply the RiemannLiouville 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, ndimentional) HurwitzLerch 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 HurwitzLerch 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 HurwitzLerch 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 HurwitzLerch Zeta function Φ(z,s,a) and some of its abovementioned generalizations, have led eventually to the following definition a family of the extended (multiparameter) HurwitzLerch Zeta functions by Srivastava et al. (2011).
Definition 1
(Srivastava et al. (2011)). The family of the extended (multiparameter) HurwitzLerch 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
and
In order to derive direct relationships of the family of the extended (multiparameter) HurwitzLerch Zeta functions
defined by (2.12) with several other relatively more familiar special functions, we need each of the following definitions.
Definition 2
The FoxWright function or , which is a further generalization of the familiar generalized hypergeometric function , with 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 Hfunction (InayatHussain 1987b, p. 4126) (see also (Buschman and Srivastava 1990) and (InayatHussain 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 is a MellinBarnes type contour starting at the point τ − i∞ and terminating at the point τ + i∞ 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).
Remark 1
If we set
and
then (2.12) reduces to the following generalized Mseries 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 socalled generalized Mseries is indeed an obvious special case of the FoxWright function defined by (2.15) (see also (Saxana 2009)). Similarly, for the generalized MittagLeffler function considered by Kilbas et al. (2002), we have
Remark 2
The following function representation can be applied in order to derive various properties of the extended HurwitzLerch Zeta function
from those of the function (see, for details, Srivastava et al. ((2011), p. 504, Theorem 8)):
the path of integration in the last member of (2.23) being a MellinBarnes type contour in the complex ξplane, 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 fractionalcalculus 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 HurwitzLerch Zeta function
In this section, we first introduce the following generating functions and their partial sums involving the extended HurwitzLerch 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 of the coefficients in (2.12) is given by (2.13). This last generating function (3.2) would reduce immediately to the expansion formula (4.1) in its special case when λ = s. Furthermore, in its limit case when
the generating function (3.2) yields
where the sequence 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 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 abovedefined 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 the second member of the assertion (3.10) of Theorem 1. Then, upon expanding the functions and _{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 wellknown result:
we get
which, in light of the definitions (2.12) and (3.2), yields the lefthand 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 HurwitzLerch 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 HurwitzLerch 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 Betafunction 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 of the coefficients given by the definition (2.13), each of the following Eulerian Betafunction integral formulas holds true:
and
provided that both sides of each of the assertions (3.21) and (3.22) exist, the FoxWright function 00 in (3.21) being tacitly interpreted as an Hfunction 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 Betafunction 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 BinSaad (2007), Gupta and Kumari (2011) and Saxena et al. (2011a).
The Eulerian Gammafunction 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 wellknown formula (3.13).
Theorem 4
Let the function be defined by (2.5). Then, in terms of the sequence of the coefficients given by the definition (2.13), each of the following single or double Eulerian Gammafunction integral formulas holds true:
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 (BinSaad 2007), (Gupta and Kumari 2011) and (Saxena et al. (2011a)).
Remark 7
Two of the claimed integral formulas in BinSaad’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 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 Gammafunction 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 of the coefficients given by the definition (2.13), each of the following Eulerian Gammafunction integral formulas holds true:
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 (BinSaad 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) BinSaad ( 2007, p. 44, Theorem 4.2):
where the sequence of the coefficients is given by the definition (2.13) and it is 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.
τGeneralizations of the HurwitzLerch Zeta functions
In a recent paper, Saxena et al. (2011a) considered a socalled τgeneralization of the HurwitzLerch 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 τgeneralization of the extended HurwitzLerch Zeta function in (2.5) as follows:
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) are no more general than their corresponding wellknown cases when 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 socalled τgeneralized functions Φ(τ;z,s,a) and can be derived simply from the corresponding (usually known) results involving the familiar functions Φ(z,s,a) and , respectively. Just for illustration of the triviality associated with such straightforward derivations, we recall the following sumintegral representation formula due to Lin and Srivastava ((2004), p. 729, Equation (20)) (see also Srivastava 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 sumintegral representation for the generalized HurwitzLerch Zeta function involved in (2.5):
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 τgeneralized HurwitzLerch Zeta function defined by (4.2), we simply make the following parameter and variable changes:
and multiply the resulting equation by τ^{−s}. By using the connection in (4.5), we thus find immediately that
In its particular case when k = 1, this last formula (4.9) would simplify at once to the following form given by Saxena et al. ((2011a), p. 311, Equation (2.2)):
which obviously is equivalent to (and certainly not a generalization of) of the τ = 1 case derived earlier by Goyal and Laddha ((1997), p. 100, Equation (1.6)).
Remark 9
The socalled τgeneralizations and of the Gauss hypergeometric function _{2}F_{1} and Kummer’s confluent hypergeometric function _{1}F_{1}, respectively, which were used in the aforecited paper by Saxena et al. ((2011a), p. 315), are obviously very specialized cases of the wellknown and extensivelyinvestigated FoxWright function _{p}Ψ_{q} defined by (2.15). In fact, it is easily seen from Definition 2 that [Saxena et al. ((2011a), pp. 315 and 317)] (see also (AlZamel 2001), (Ali et al.2001) and (Virchenko et al.2001))
and
Similar remarks and observations would apply equally strongly to the other τgeneralizations of wellknown and extensivelyinvestigated hypergeometric functions in one, two and more variables.
We conclude this section by presenting a generalization of the sumintegral representation formula (4.6) due to Lin and Srivastava ((2004), p. 729, Equation (20)).
Theorem 6
The following sumintegral representation formula holds true:
provided that each member of the assertion (4.13) exists.
Proof
First of all, in light of the following elementary series identity:
we find from the definition (2.12) that
The assertion (4.13) of Theorem 6 would now emerge readily upon first appealing to the aforementioned known result due to Srivastava et al. ((2011), p. 504, Equation (6.4)) (see also Remark 3 above) given by
and then setting
□
Obviously, in its special case when
and
the general result (4.13) asserted by Theorem 6 would reduce immediately to the known sumintegral representation formula (4.6) due to Lin and Srivastava ((2004), p. 729, Equation (20)).
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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