Note on the Hurwitz–Lerch Zeta Function of Two Variables

: A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

Here, in a systematic way, we aim to establish certain formulas and representations for the extended Hurwitz-Lerch zeta function of two variables (6) such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to produce corresponding results for several less generalized Hurwitz-Lerch zeta functions than the extended Hurwitz-Lerch zeta function (6). Further, two more generalized settings than (6) are provided.
The following confluent form of the Appell hypergeometric function F 1 is recalled (see, e.g., [1] p. 225, Equation (21)); see also ([21] p. 22 et seq.) We provide further integral representations of the extended Hurwitz-Lerch zeta function (6), asserted in the following theorem. Theorem 1. Each of the following integral representations holds.

Generating Functions of (6)
Certain generating functions for the extended Hurwitz-Lerch zeta function (6) are given in the following theorem. Theorem 2. The following two formulas hold true: and Proof. We begin by recalling the generalized binomial theorem where λ ∈ C and |y| < 1.
We can also prove the generating relations here by using some known generating relations for F 1 (see [29] Equation (2.1)) and ( [22] Equations (1.2)-(1.3)) in (9). The details of the proof are omitted here. (6) Certain derivative formulas for the extended Hurwitz-Lerch zeta function (6) are established in the following theorem.

Derivative Formulas of
Theorem 3. Each of the following derivative formulas holds true for m, n ∈ N 0 : Proof. Differentiating the series definition (6) with respect to the variable x under the double summations, which is valid under the conditions in (6), we have Putting k − 1 = k in (35) and cancelling the prime on k, we obtain Differentiating the right side of (37), successively, m − 1 times, with respect to the variable x, we have (32). Similarly, we can obtain (33) and (34). The details are omitted. (6) Wang [30] presented a number of recurrence relations for F 1 , which are chosen to give some recurrence relations for the extended Hurwitz-Lerch zeta function (6), asserted in Theorem 4. Theorem 4. Let n ∈ N 0 . Then the following recurrence relations are satisfied:

Recurrence Relations of
Here the involved empty sum in each identity is assumed to be nil.
Using six known recurrence relations for 1 F 1 in (12), we establish six recurrence relations for the extended Hurwitz-Lerch zeta function (6), which are asserted in Theorem 5.

Symmetries and Conclusions
We can find some interesting identities from symmetries involved in certain definitions and formulas. From (6) and (10), we demonstrate the follow symmetric relations: and Further, in view of the symmetric relation (48), each integral representation in Theorem 1 may yield another integral representation. For example, from (14) and (48), we have where min{ (s), (α), (a)} > 0 and max{ (x), (y)} < 1.
The extended Hurwitz-Lerch zeta function of two variables in (6) may be further generalized in various ways. Here we introduce two extensions, one of which is due to parametric increase and the other of which is due to variable addition: (a) m 1 +···+m n (b 1 ) m 1 · · · (b n ) m n (c) m 1 +···+m n m 1 ! · · · m n ! x m 1 1 · · · x m n n (m 1 + · · · + m n + α) s .
The extended Hurwitz-Lerch zeta function (6) can be specialized to yield several known generalizations of the Hurwitz-Lerch zeta function (5) (see, e.g., [20]). Thus, the results presented here can yield corresponding identities regarding several reduced cases of the extended Hurwitz-Lerch zeta function (6), which are still generalizations of the Hurwitz-Lerch zeta function (5).
Author Contributions: These authors contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported by the Dongguk University Research Fund of 2020.