1. Introduction, Definitions and Preliminaries

We begin by recalling the familiar general Hurwitz-Lerch Zeta function , which is defined by (see, for example, ^[2]; see also ^{[18, 19, 20, 21, 22]}).

(1.1)

(, when ; when ).

Special cases of the Hurwitz-Lerch Zeta function include (for example) the Riemann Zeta function and the Hurwitz (or generalized) Zeta function defined by (see, for details, [2, Chapter I] and [21, Chapter 2])

(1.2)

and

(1.3)

respectively. Just as its aforementioned special cases and , the Hurwitz-Lerch Zeta function defined by (1.4) 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 [^[2], Equation 1.11 (3)]

(1.4)

(, when ; when ).

Recently, the following modified (and slightly generalized) version of the integral in (1.4) was introduced and studied by Raina and Chhajed [^[16], Equation (1.6)]:

(1.5)

where we have assumed further that

or

provided, of course, that the integral in (1.5) exists. As a matter of fact, the aforementioned investigation by Raina and Chhajed ^[16] was motivated by the following special case of the function defined by (1.5):

(1.6)

(, when ; when ).

where the function defined by

(1.7)

was studied by Goyal and Laddha [^[4], Equation (1.5)]. Here, and in what follows, denotes the Pochhammer symbol (or the shifted factorial) which is defined, in terms of the familiar Gamma function, by

where it is understood conventionally that and assumed tacitly that the G-quotient exists, being the set of positive integers.

It may be of interest to observe in passing that, in terms of the Riemann-Liouville fractional derivative operator defined by (see, for example, ^{[3, 7, 17]})

the series definitions in (1.1) and (1.7) readily yield

(1.8)

which (as already remarked by Lin and Srivastava ^[8]) exhibits the interesting (and useful) fact that the function is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz-Lerch function .

One other special case of the function defined by (1.5) occurs when we set and in the definition (1.5). We thus obtain

(1.9)

where is the extended Hurwitz zeta function defined in ^[1]. In fact, just as it is already pointed out in ^[10], the series representation (see [^[16], Equation (2.1)]) given for the function in (1.5) is incorrect. Obvious further specializations in (1.6) and (1.9) would immediately relate these functions with the Riemann zeta function and the Hurwitz (or generalized) zeta function defined by (1.2) and (1.3), respectively.

By using the series expansion of the binomial occurring in the integrand of (1.5) and evaluating the resulting integral by means of the corrected version of a known integral formula [^[13], Equation (1.53)] in terms of Fox’s H-function defined by (1.12) below, the following series and Mellin-Barnes type contour integral representations of the function defined by (1.5) were obtained in ^[10]:

(1.10)

and

(1.11)

it being assumed that each member of the assertions (1.10) and (1.11) exists.

Remark 1. The H-functions involved on the right-hand sides of (1.10) and (1.11) are particular cases of the celebrated Fox’s H-function which is defined as follows.

Definition 1. The well-known Fox’s H-function is defined here by (see, for details, [^[13], Defintion 1.1]; see also [^{[6, 11, 23, 24]})

(1.12)

where

(1.13)

Here

an empty product is interpreted as 1, m, n, p and q are integers such that and ,

and is a suitable Mellin-Barnes type contour separating the poles of the gamma functions

from the poles of the gamma functions

In our present investigation, we consider certain statistical applications of the generalized Hurwitz-Lerch zeta function defined by (1.5). We first derive a partial differential equation satisfied by the function in (1.11). We then obtain another series representation and related results for this generalized Hurwitz-Lerch zeta function. The results derived here are also applied in our investigations concerning the generalized Hurwitz-Lerch zeta measure and its related statistical concepts.

2. Differential Equation of the Generalized Hurwitz-Lerch Zeta Function

In this section, we will show that the generalized Hurwitz-Lerch zeta function satisfies a partial differential equation when the parameter is given by

We first prove the following lemma which will be used in the proof of our main theorem.

Lemma (Derivative Property). The following derivative formulas hold true:

(2.1)

and

(2.2)

Proof. The proofs of the derivative formulas (2.1) and (2.2) are direct. For example, by applying the series representation (1.10), one easily finds that

which is precisely the first result (2.1) asserted by the Lemma. The second assertion (2.2) follows immediately from (2.1) upon setting

Our first main result is contained in the following theorem.

Theorem 1. The generalized Hurwitz-Lerch zeta function satisfies the following partial differential equation:

(2.3)

where the differential operators and are given by

(2.4)

and

respectively.

Proof. We first rewrite the H-function occurring in the Mellin-Barnes type contour integral representation (1.11) as follows:

(2.5)

where is a suitable Mellin-Barnes type contour integral in the complex w-plane. By setting

in the above equation (2.5) and then applying the well-known (Gauss-Legendre) multiplication formula:

(2.6)

we find that

(2.7)

where

is a very specialized case of Meijer’s G-function which, in turn, is a special case of Fox’s H-function defined by (1.12), that is, we have the following relationship (see, for details, ^[14]; see also ^{[2, 12, 15]}):

(2.8)

We know that the function defined by

satisfies the following differential equation of order (see, for example, [^[2], Equation 5.4(1)]):

where

Hence, clearly, the function given by (2.7) satisfies the following differential equation:

(2.9)

where, as already stated in Theorem 1,

Now, if we write [see also Equation (2.4)]

then the equation (2.9) becomes

(2.10)

Applying the differential operator to the function given by (1.11) with

we find by using (2.10) that

(2.11)

where the first integral is actually the generalized Hurwitz-Lerch zeta function given by

(2.12)

The evaluation of the second integral given by

(2.13)

is more complicated. Since the residues of at the poles are computed by

(2.14)

the Residue Theorem implies that

(2.15)

Applying (2.2) in (2.15), we get

(2.16)

Now, upon substituting from (2.12) and (2.16) into (2.11), we obtain

(2.17)

which, after a little simplification, becomes

(2.18)

Finally, by setting

in the last equation (2.18), we arrive at the desired result (2.3) asserted by Theorem 1.

It is interesting to consider a special case of Theorem 1 when m = 1. Thus, if we write

(2.19)

then we have the following corollary.

Corollary 1. The generalized Hurwitz-Lerch zeta function satisfies the following partial differential equation:

(2.20)

Furthermore; the function , considered as an analytic function of the variable b satisfies the following relation:

(2.21)

3 Further Series Representations and Related Results

In this section, we first give a new series representation of the generalized Hurwitz-Lerch zeta function involving the familiar Laguerre polynomials of order (index) α and degree in , which are generated by

(3.1)

Indeed, upon setting

in (3.1), we get

(3.2)

We now make use of (3.2) and the series expansions of the binomials

occurring in the integrand of (1.5). By evaluating the resulting Eulerian integral, we thus arrive at the series representations given by Theorem 2 below.

Theorem 2. Each of the following series representations holds true for the generalized Hurwitz-Lerch zeta function :

(3.3)

and

(3.4)

provided that each member of the assertions (3.3) and (3.4) exists, being given by (1.7).

Proof. As already outlined above, our demonstration of the first assertion (3.3) of Theorem 2 is based essentially upon the representation (3.2) and the following well-known Eulerian integral:

(3.5)

The second assertion (3.4) follows from the first assertion (3.3) when we interpret the -series in (3.4) by means of the definition (1.7).

In our derivation of each of the summation formulas (3.3) and (3.4), it is assumed that the required inversions of the order of summation and integration are justified by absolute and uniform convergence of the series and integrals involved. The final results (3.3) and (3.4) would thus hold true whenever each member of the assertions (3.3) and (3.4) of Theorem 2 exists.

Remark 2. For the extended Hurwitz zeta function defined by (1.9), it is easily deduced from the assertion (3.4) of Theorem 2 when and that

(3.6)

provided that each member of (3.6) exists, being the Hurwitz (or generalized) zeta function given by (1.3). The obvious further special case of (3.6) when and would yield the corrected version of a known result (see [^[1], Equation (7.78)]).

We now give a pair of summation formulas involving the generalized Hurwitz-Lerch zeta function

Theorem 3. Each of the following summation formulas holds true for the generalized Hurwitz-Lerch zeta function :

(3.7)

and

(3.8)

provided that each member of the assertions (3.7) and (3.8) exists.

Proof. Making use of the integral representation in (1.5) for the function , we get

(3.9)

Since

(3.10)

by substituting from (3.10) into (3.9) and interchanging the order of summation and integration, we find that

(3.11)

which obviously proves the assertion (3.7) of Theorem 3.

The assertion (3.8) of Theorem 3 can be proven in a manner analogous to that detailed above.

Remark 3. If we set in (3.7) and (3.8), the series occurring on their right-hand sides would terminate. Upon setting and , we thus obtain

(3.12)

and

(3.13)

In particular, if we set in these last two summation formulas (3.12) and (3.13), we get

(3.14)

and

(3.15)

respectively. In its further special case when , the summation formula (3.14)can be shown to correspond to known results (see, for example, [^[1], Theorem 7.9]; see also ^[1]).

4. A Generalization of the Hurwitz Measure

Suppose that denote the characteristic function of the subset of the set of positive integers (or, in the language of probability theory, the indicator function of the event ). Then it is well known that the following arithmetic density of number theory:

(4.1)

does not define a measure on the set of positive integers. In order to remedy this deficiency, Golomb ^[5] defined a probability on the sample space as follows:

(4.2)

where denotes the Riemann zeta function defined by (1.2) and the characteristic (or indicator) function is given by

(4.3)

Furthermore, Golomb ^[5] showed that, if the subset A of has an arithmetic density, then

(4.4)

thereby allowing number-theoretic facts regarding densities of sets of positive integers to be proven by probabilistic means and then showing that such properties are preserved in the limit.

In an interesting sequel to Golomb’s investigation ^[5], Lippert ^[9] gave an analogous definition of the probabilities when the set is replaced by the set of all real numbers greater than 1. Thus, for a Borel set , Lippert’s Hurwitz measure of the set is defined by (see, for details, [^[9], Definition 1]

(4.5)

or, equivalently, by

(4.6)

where, in terms of the Hurwitz (or generalized) zeta function defined by (1.3),

(4.7)

In this section, we propose to introduce a new continuous analogue of Lippert’s Hurwitz measure in (4.5) by using a special case of the generalized Hurwitz-Lerch zeta function defined by (1.5).

Definition 2. For a Borel set , the generalized Hurwitz measure of the set is defined by

(4.8)

or, equivalently, by

(4.9)

where

(4.10)

and

(4.11)

since it is easily seen from the definition (1.5) that

(4.12)

In view of the following relationship:

the generalized Hurwitz measure in (4.8) or (4.9) also defines a probability measure on .

Remark 4. For and by letting we have

(4.13)

which implies that

(4.14)

Thus, clearly, can be continuously approximated by .

Proposition. The measure satisfies the following difference equation:

(4.15)

Proof. From the series representation (1.10) of (with and ), we have

(4.16)

The difference equation (4.15) now follows on combining (4.10) and (4.16).

Remark 5. For and by letting , the difference equation (4.15) reduces to the following form:

(4.17)

where is given by (4.7).

For open events, the generalized Hurwitz measure in (4.8) or (4.9) can be evaluated by using (4.9) and the above Proposition. The results are being stated as Theorem 4 below.

Theorem 4. If then

(4.18)

More generally; the generalized Hurwitz measure of an open set is given by

(4.19)

where

The following theorem shows that the generalized Hurwitz measure in (4.8) or (4.9) basically inherits all properties of Lippert’s Hurwitz measure given by (4.5) or (4.6).

Theorem 5. Corresponding to the generalized Hurwitz measure given by (4.19), let

(4.20)

Then

(4.21)

Proof. From (4.19), we have

(4.22)

By expanding the function by means of Taylor’s series and using the derivative formula (4.12), we get

(4.23)

We now consider each sum in (4.23) separately. We thus find that

(4.24)

Since the number of non-negative integer solutions of the Diophantine equation is

the double summation in (4.24) can be replaced by a single summation, that is,

(4.25)

We thus obtain

(4.26)

We note that, when , the series for is divergent and the series for is convergent. Therefore, all other terms vanish in (4.26) except the leading term. Consequently, we get

(4.27)

which completes the proof of Theorem 5.

As in the theory of probability, we introduce the following definition.

Definition 3. A random variable is said to be generalized Hurwitz distributed if its probability density function (p.d.f.) is given by

(4.28)

Theorem 6. Let be a continuous random variable with its p.d.f defined by (4.28). Then the moment generating function of the random variable is given by

(4.29)

with the moment of order n given by

(4.30)

Proof. The assertion in (4.29) follows easily by using the exponential series for . If we use integration by parts, we find from the definition that

(4.31)

where we have also used the derivative property (4.12) and the following limit formula:

(4.32)

Consequently, we have the following reduction formula for :

(4.33)

By iterating the recurrence (4.33), we arrive at the desired result (4.30) asserted by Theorem 6.

Remark 6. The assertion (4.30) of Theorem 6 provides a generalization of a known result [^[9], Proposition 3].

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