Certain Identities of a General Class of Hurwitz-Lerch Zeta Function of Two Variables

In this paper, we introduce a generalized double Hurwitz-Lerch Zeta function and then systematically investigate its properties and various integral representations. Further, we show that these results provide certain known as well as new extensions of earlier stated results of generalized Hurwitz-Lerch Zeta functions.

It is known that η(s, a) and so also η(s) can be continued analytically to the whole complex s -plane. Therefore, each of the functions η(s, a) and η(s) is an entire function of s, a ∈ C.

The General Hurwitz-Lerch Zeta Function of Two Variables and Related Double Zeta Functions
In this section, we define a general Hurwitz-Lerch Zeta function of two variables consisting of a double sequence of arbitrary complex numbers and then making an appeal to the convergence conditions of double series in ( [13], [14], [22], [23], [24], and [25]), we obtain convergence conditions for general Hurwitz-Lerch Zeta function of two variables and its related double zeta functions.
Finally, we are led to the assertion of Theorem 2.1.

Example 2.2
Let m > 0, n ≥ 0 and A m,n = The limit in (17) exists if , ε > 0 and using the concept of Geometric Mean < Harmonic Mean).
Hence by (15) and (19), we find that Similarly, we get Since for special values, a double sequence of arbitrary complex numbers {A m,n } reduces to all the extended generalized Hurwitz-Lerch zeta functions defined in Section 1 and other known zeta functions studied in the work due to ( [6] and [3]), we shall call it the unified Hurwitz-Lerch zeta function of two variables.
Proof. Following the methods used to prove Theorem 2.1, we can evaluate existence conditions of (26) and prove the Theorem 2.2.

(47)
Result 5.1. For all |z| + |t| < 1, and a, α 1 ∈ C, β 1 ∈ C\Z − 0 , R(a) > 1, (α 1 ) > 0, there exists an integral representation φ α 1 ,α 2 ,1;β 1 ,2 (z, t, s, a) = 1 α 1 tΓ(s) Additionally, it may be shown that an integral representation of Appell's F 2 (.) series can be used to generate many integral representations like (48),(49) in terms of the hypergeometric functions 2 F 1 and 3 F 2 which are very useful in applications. A listing of some useful reduction (and transformation) formulas is provided in [16]. As a consequence of the present work, special cases of the integrals containing the Appell hypergeometric function F 2 (.) can now be expressed in terms of elementary hypergeometric and algebraic functions.