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

Hari M

Author Affiliations

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

SpringerPlus 2013, 2:67  doi:10.1186/2193-1801-2-67


The electronic version of this article is the complete one and can be found online at: http://www.springerplus.com/content/2/1/67


Received:20 November 2012
Accepted:4 January 2013
Published:25 February 2013

© 2013 Srivastava; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M1">View MathML</a>-function; Mittag-Leffler type functions; Mellin-Barnes type integral representations and Meromorphic continuation; Generating functions and Eulerian Gamma-function and Beta-function integral representations

Introduction and preliminaries

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M2">View MathML</a>

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M3">View MathML</a>

Here, as usual,<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M4">View MathML</a> denotes the set of integers,<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M5">View MathML</a> denotes the set of real numbers, <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M6">View MathML</a> denotes the set of positive real numbers and<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M7">View MathML</a> denotes the set of complex numbers.

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.)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M8">View MathML</a>

(1.1)

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 s(ξ) defined by (see, for details, (Erdélyi et al.1953, Chapter I) and Srivastava and Choi ((2001), Chapter 2)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M9">View MathML</a>

(1.2)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M10">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M11">View MathML</a>

(1.4)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M12">View MathML</a>

(1.5)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M13">View MathML</a>

(1.6)

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))

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M14">View MathML</a>

(1.7)

Making use of the Pochhammer symbol (or the shifted factorial) <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M15">View MathML</a> defined, in terms of the familiar Gamma function, by

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M16">View MathML</a>

(1.8)

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:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M17">View MathML</a>

(1.9)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M18">View MathML</a>

(1.10)

More generally, it is not difficult to show similarly that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M19">View MathML</a>

(1.11)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M20">View MathML</a>

this last result (1.11) yields

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M21">View MathML</a>

(1.12)

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 ϑλ(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:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M22">View MathML</a>

(1.13)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M23">View MathML</a>

(1.14)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M24">View MathML</a>

(1.15)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M25">View MathML</a>

(1.16)

so that, obviously, we have

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M26">View MathML</a>

(1.17)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M27">View MathML</a>

(1.18)

For the Riemann Zeta function ζ(s), the special case of each of the generating functions ϑλ(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 ϑλ(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 ϑλ(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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M28">View MathML</a>

(2.1)

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)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M29">View MathML</a>

(2.2)

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)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M30">View MathML</a>

(2.3)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M31">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M32">View MathML</a>

(2.5)

where, as already pointed out by Lin and Srivastava (2004), <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M33">View MathML</a> 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,s,a) and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M34">View MathML</a> was studied, in the following form, by Garg et al.((2008), p. 313, Equation (1.7)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M35">View MathML</a>

(2.6)

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(z,s,a) studied by Garg et al.(2008) does not provide a generalization of the function <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M36">View MathML</a> which was introduced earlier by Lin and Srivastava (2004). Indeed, for λ = 1, the function Φλ,μ;ν(z,s,a) coincides with a special case of the function <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M37">View MathML</a> when ρ = σ = 1, that is,

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M38">View MathML</a>

Next, for the Riemann-Liouville fractional derivative operator<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M39">View MathML</a> defined by (see, for example, Erdélyi et al.((1954), p. 181), Samko et al.(1993) and (Kilbas et al.2006, p. 70 et seq.))

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M40">View MathML</a>

(2.7)

the following formula is well-known:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M41">View MathML</a>

(2.8)

which, by virtue of the definitions (1.1) and (2.3), yields the following fractional derivative formula for the generalized Hurwitz-Lerch Zeta function <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M42">View MathML</a>withρ = σ [Lin and Srivastava ((2004), p. 730, Equation (24))]:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M43">View MathML</a>

(2.9)

In its particular case when ν = σ = 1, the fractional derivative formula (2.9) would reduce at once to the following form:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M44">View MathML</a>

(2.10)

which (as already remarked by Lin and Srivastava (2004), p. 730) exhibits the interesting (and useful) fact that <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M45">View MathML</a> is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz-Lerch function Φ(z,s,a). Moreover, it is easily deduced from the fractional derivative formula (2.8) that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M46">View MathML</a>

(2.11)

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M47">View MathML</a>two times as indicated above in (2.11). The interested reader may be referred also to many other explicit representations for <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M48">View MathML</a> and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M49">View MathML</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, n-dimentional) Hurwitz-Lerch Zeta function Φn(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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M50">View MathML</a>

is defined by

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M51">View MathML</a>

(2.12)

where the sequence <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M52">View MathML</a> of the coefficients in the definition (2.12) is given, for latter convenience, by

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M53">View MathML</a>

(2.13)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M54">View MathML</a> denotes the Pochhammer symbol given by (1.8) and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M55">View MathML</a>

(2.14)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M56">View MathML</a>

In order to derive direct relationships of the family of the extended (multi-parameter) Hurwitz-Lerch Zeta functions

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M57">View MathML</a>

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M58">View MathML</a> or <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M59">View MathML</a>, which is a further generalization of the familiar generalized hypergeometric function <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M60">View MathML</a>, with p numerator parameters a1, ⋯,ap and q denominator parameters b1, ⋯,bq such that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M62">View MathML</a>

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))

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M63">View MathML</a>

(2.15)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M64">View MathML</a>

where the equality in the convergence condition holds true for suitably bounded values of |z| given by

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M65">View MathML</a>

(2.16)

In the particular case when

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M66">View MathML</a>

we have the following relationship (see, for details, (Choi et al.1985, p. 21)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M67">View MathML</a>

(2.17)

in terms of the generalized hypergeometric function <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M68">View MathML</a>.

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)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M69">View MathML</a>

(2.18)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M70">View MathML</a>

which contains fractional powers of some of the Gamma functions involved. Here, and in what follows, the parameters

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M71">View MathML</a>

the exponents

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M72">View MathML</a>

can take on noninteger values, and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M73">View MathML</a> is a Mellin-Barnes type contour starting at the point τ − i and terminating at the point τ + i<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M74">View MathML</a> 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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M75">View MathML</a>

(2.19)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M76">View MathML</a>

(2.20)

where Λ is defined by (2.19).

Remark 1

If we set

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M77">View MathML</a>

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M78">View MathML</a>

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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M79">View MathML</a>

(2.21)

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M80">View MathML</a> defined by (2.15) (see also (Saxana 2009)). Similarly, for the generalized Mittag-Leffler function considered by Kilbas et al. (2002), we have

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M81">View MathML</a>

(2.22)

Remark 2

The following <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M82">View MathML</a>-function representation can be applied in order to derive various properties of the extended Hurwitz-Lerch Zeta function

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M83">View MathML</a>

from those of the <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M84">View MathML</a>-function (see, for details, Srivastava et al. ((2011), p. 504, Theorem 8)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M85">View MathML</a>

(2.23)

the path of integration<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M86">View MathML</a> in the last member of (2.23) being a Mellin-Barnes type contour in the complex 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(λj+ρjξ) (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))]:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M87">View MathML</a>

(2.24)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M88">View MathML</a>

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M89">View MathML</a>

(3.1)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M90">View MathML</a>

(3.2)

where the sequence <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M91">View MathML</a> 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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M92">View MathML</a>

the generating function (3.2) yields

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M93">View MathML</a>

(3.3)

where the sequence <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M94">View MathML</a> 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 Ωλ(z,t;s,a) and Θ(z,t;s,a) in (3.2) and (3.3), respectively:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M95">View MathML</a>

(3.4)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M96">View MathML</a>

(3.5)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M97">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M98">View MathML</a>

(3.7)

which obviously satisfy the following decomposition formulas:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M99">View MathML</a>

(3.8)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M100">View MathML</a>

(3.9)

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:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M101">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M102">View MathML</a>

(3.11)

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

Proof

For convenience, we denote by<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M103">View MathML</a> the second member of the assertion (3.10) of Theorem 1. Then, upon expanding the functions <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M104">View MathML</a> and 1F1 in series forms, we find that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M105">View MathML</a>

(3.12)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M106">View MathML</a>

(3.13)

we get

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M107">View MathML</a>

(3.14)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M108">View MathML</a>

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)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M109">View MathML</a>

(3.15)

or, equivalently,

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M110">View MathML</a>

(3.16)

Theorem 2

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M111">View MathML</a>

(3.17)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M112">View MathML</a>

(3.18)

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:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M113">View MathML</a>

(3.19)

Furthermore, in their special case when

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M114">View MathML</a>

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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M115">View MathML</a>

(3.20)

we evaluate several Eulerian Beta-function integrals involving the generating functions Ωλ(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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M116">View MathML</a> of the coefficients given by the definition (2.13), each of the following Eulerian Beta-function integral formulas holds true:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M117">View MathML</a>

(3.21)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M118">View MathML</a>

(3.22)

provided that both sides of each of the assertions (3.21) and (3.22) exist, the Fox-Wright function 00<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M119">View MathML</a> in (3.21) being tacitly interpreted as an 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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M120">View MathML</a>

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 Ωλ(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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M121">View MathML</a> be defined by (2.5). Then, in terms of the sequence <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M122">View MathML</a> of the coefficients given by the definition (2.13), each of the following single or double Eulerian Gamma-function integral formulas holds true:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M123">View MathML</a>

(3.23)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M124">View MathML</a>

(3.24)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M125">View MathML</a>

(3.25)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M126">View MathML</a>

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 Ωλ(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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M127">View MathML</a> of the coefficients given by the definition (2.13), each of the following Eulerian Gamma-function integral formulas holds true:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M128">View MathML</a>

(3.26)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M129">View MathML</a>

(3.27)

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M130">View MathML</a>

(3.28)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M131">View MathML</a>

(3.29)

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

Remark 8

Several specialized cases of Theorem 5 when

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M132">View MathML</a>

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 Ωλ(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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M133">View MathML</a>

(3.30)

where the sequence <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M134">View MathML</a> 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.

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))]:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M135">View MathML</a>

(4.1)

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M136">View MathML</a> in (2.5) as follows:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M137">View MathML</a>

(4.2)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M138">View MathML</a>

(4.3)

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:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M139">View MathML</a>

(4.4)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M140">View MathML</a>

(4.5)

Clearly, therefore, the definitions in (4.1) and (4.2) (with<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M141">View MathML</a>) are no more general than their corresponding well-known 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 so-called τ-generalized functions Φ(τ;z,s,a) and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M142">View MathML</a> can be derived simply from the corresponding (usually known) results involving the familiar functions Φ(z,s,a) and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M143">View MathML</a>, respectively. Just for illustration of the triviality associated with such straightforward derivations, we recall the following sum-integral 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):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M144">View MathML</a>

(4.6)

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M145">View MathML</a> involved in (2.5):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M146">View MathML</a>

(4.7)

or, equivalently,

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M147">View MathML</a>

(4.8)

where, for convenience, Δ(n;λ) abbreviates the array of n parameters

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M148">View MathML</a>

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 <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M149">View MathML</a> defined by (4.2), we simply make the following parameter and variable changes:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M150">View MathML</a>

and multiply the resulting equation by τs. By using the connection in (4.5), we thus find immediately that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M151">View MathML</a>

(4.9)

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)):

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M152">View MathML</a>

(4.10)

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 so-called τ-generalizations <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M153">View MathML</a> and <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M154">View MathML</a> of the Gauss hypergeometric function 2F1 and Kummer’s confluent hypergeometric function 1F1, respectively, which were used in the aforecited paper by Saxena et al. ((2011a), p. 315), are obviously very specialized cases of the well-known and extensively-investigated Fox-Wright function pΨq defined by (2.15). In fact, it is easily seen from Definition 2 that [Saxena et al. ((2011a), pp. 315 and 317)] (see also (Al-Zamel 2001), (Ali et al.2001) and (Virchenko et al.2001))

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M155">View MathML</a>

(4.11)

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M156">View MathML</a>

(4.12)

Similar remarks and observations would apply equally strongly to the other τ-generalizations of well-known and extensively-investigated hypergeometric functions in one, two and more variables.

We conclude this section by presenting a generalization of the sum-integral representation formula (4.6) due to Lin and Srivastava ((2004), p. 729, Equation (20)).

Theorem 6

The following sum-integral representation formula holds true:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M157">View MathML</a>

(4.13)

provided that each member of the assertion (4.13) exists.

Proof

First of all, in light of the following elementary series identity:

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M158">View MathML</a>

we find from the definition (2.12) that

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M159">View MathML</a>

(4.14)

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

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M160">View MathML</a>

(4.15)

and then setting

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M161">View MathML</a>

Obviously, in its special case when

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M162">View MathML</a>

and

<a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M163">View MathML</a>

the general result (4.13) asserted by Theorem 6 would reduce immediately to the known sum-integral 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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