Further Generalization of the Extended Hurwitz-Lerch Zeta Functions

abstract: Recently various extensions of Hurwitz-Lerch Zeta functions have been investigated. Here, we first introduce a further generalization of the extended Hurwitz-Lerch Zeta functions. Then we investigate certain interesting and (potentially) useful properties, systematically, of the generalization of the extended Hurwitz-Lerch Zeta functions, for example, various integral representations, Mellin transform, generating functions and extended fractional derivatives formulas associated with these extended generalized Hurwitz-Lerch Zeta functions. An application to probability distributions is further considered. Some interesting special cases of our main results are also pointed out.


Introduction, Definitions and Preliminaries
Throughout this paper, let C, Z − 0 , and N denote the sets of complex numbers, nonpositive integers, and positive integers, respectively, and N 0 := N ∪ {0}.The Hurwitz-Lerch Zeta function Φ(z, s, a) is defined by (see, e.g., [5, p. 27, Eq. 1.11 (1)]; see also [16, p. 121] and [17, p. 194 Various generalizations of the Hurwitz-Lerch Zeta function Φ(z, s, a) have been considered by many authors (see, e.g., [3,4,5,11,16,17,18]).For example, Goyal and Laddha [7, p.  Very recently, Parmar and Raina [14, p. 120, Eq. (2.1)] introduced the following extension of generalized Hurwitz-Lerch Zeta function: and Clearly, the special case of (1.6) when p = 0 reduces immediately to (1.3).Further, Özergin et al. [13] introduced the following generalizations of the extended Beta and hypergeometric functions which are defined, respectively, by Evidently, the special cases of (1.10) and (1.11) when ρ = σ reduce, respectively, to the extended Beta and hypergeometric functions (1.8) and (1.9).Motivated mainly by various recent interesting extensions of the Hurwitz-Lerch Zeta function, we introduce a further generalization of the extended Hurwitz-Lerch Zeta functions and investigate its certain interesting and (potentially) useful properties such as various integral representations, Mellin transform, generating functions, derivative formulas and relations associated with extended fractional derivative operator.An application to probability distributions is further considered.Some interesting special cases of our main results are also indicated.

Integral Representations
We present certain integral representations for the extended Hurwitz-Lerch Zeta function in (2.1).
Theorem 3.1.The following integral representation holds true: Proof: Using the Eulerian integral of the Gamma function Γ(s) (see, e.g., [17, p. 1, Eq. ( 1)]), it is easy to find the following identity: Applying (3.2) to (2.1) and interchanging the order of summation and integration which may be valid under the conditions stated in Theorem 3.1, we get Finally the use of (1.11) to the last expression is seen to lead to the desired result.✷ Theorem 3.2.The following integral representations hold true: , s, a dx provided the integrals in the right-hand sides of (3.3) and (3.4) converge.
Proof: Using the integral representation of the Pochhammer symbol (λ) n : in (2.1) and inverting the order of summation and integration which may be permissible under the conditions stated Theorem 3.3, we get  [14].Further for p = 0, we obtain corresponding results in [6].

Mellin Transform and Generating Relations
The Mellin transform of a suitable integrable function f (t) with index α is defined, as usual, by provided the improper integral in (4.1) exists.

183
Proof: Taking the Mellin transform (4.1) for (2.1), we find Applying the following known result (see [1, p. 21, Eq.(2.1)]): we obtain Using the known result [13, p. 4603, Eq.( 5)]: Proof: Let the left-hand side of the assertion (4.8) be denoted by S. Then we find from (2.1) that where the second equality follows by reversing the order of summations and using the identity (λ Now, applying the binomial expansion and considering (2.1) as a function of the form Φ Extensions of Hurwitz-Lerch Zeta Functions

185
More generally (4.11) Proof: Using (2.1) in the right-hand side of the assertion (4.9), we have Then, by making use of (2.1), we are led to the desired assertion (4.9).
Similarly, it is not difficult to show the assertion in (4.10).The generating function (4.10) would reduce immediately to the expansion formula (4.9) in its special case when δ = s.
Next, by virtue of the following limit formula: when t is replaced by t λ and | λ |→ ∞ in (4.10), we get the desired exponential generating function asserted by (4.11).✷ R. K. Parmar, J. Choi and S. D. Purohit

Extended Fractional Derivative Operator
For the Riemann-Liouville fractional derivative operator D µ z defined by (see, e.g., [15] and [10, p. 70 et seq.]): it is known that It is noted that the path of integration in the definition (5.1) is a line in the complex t-plane from t = 0 to t = z.Further, Srivastava et al. [19, p. 243 ] introduced an extended Riemann-Liouville fractional derivative operator as follows: ) where, as before, ℜ(p) ≧ 0. The path of integration in the Definition (5.3), which immediately yields the definition (5.1) when p = 0, is also a line in the complex t-plane from t = 0 to t = z.
Also when ρ = σ, (5.3) reduces to the extended Riemann-Liouville fractional derivative operator D µ,p z introduced by Özarslan and Özergin [12]: (5.4) Making use of the Definition (5.3), we can easily derive the following analogue of the familiar fractional derivative formula (5.2) which would readily yield an extension of the fractional derivative formula in Theorem 7 and Theorem 8 below.

Application to the Probability Distributions
Here we consider a general probability distribution involving the extended generalized Hurwitz-Lerch Zeta function (2.1) defined as follows: A continuous random variable ξ is said to be generalized Hurwitz distributed if its probability density function is given by where it is tacitly assumed that the arguments z, s, p and the parameters λ, µ and ν are fixed and suitably constrained so that the probability density function f ξ (a) remains nonnegative.
Theorem 6.1.The moment generating function M (t) of the continuous random variable ξ of probability density function f ξ (a) in (6.1) is given as follows: where the moments E s [ξ n ] of order n are given by

.3)
Proof: The assertion in (6.2) can be derived easily by using the series expansion of e ξt .To establish (6.3), we observe that where, in addition to the derivative property (6.4), we have used the following limit formula: Extensions of Hurwitz-Lerch Zeta Functions and by iterating the recurrence (6.5), we arrive at the desired result (6.3).✷