A Series Representation for the Hurwitz–Lerch Zeta Function

: We derive a new formula for the Hurwitz–Lerch zeta function in terms of the inﬁnite sum of the incomplete gamma function. Special cases are derived in terms of fundamental constants.


Significance Statement
In 1887 Mathias Lerch [1] produced his famous manuscript on the Hurwitz-Lerch zeta function. Series representations of the Hurwitz-Lerch zeta function have been extensively studied in [1][2][3]. The Hurwitz-Lerch zeta function generalizes the Hurwitz zeta function, the polylogarithm function, and many interesting and important special functions.
The purpose of this work is to contribute to the existing literature on Hurwitz-Lerch zeta function series representations by giving a formal derivation of the Hurwitz-Lerch zeta function expressed in terms of the infinite sum of the incomplete gamma function. We expect that researchers will find this new integral formula helpful in their present and future study. Every new finding on the Hurwitz-Lerch zeta function is significant due to the function's many applications in both applied and pure mathematics. This study contains no previously published findings.

Introduction
We develop a novel formulation for the Hurwitz-Lerch zeta function in terms of the infinite sum of the incomplete gamma functions given by where the variables k, a, m are general complex numbers and the branch cut for log(a) is given by Equation (4.2.3) in [4]. This new expression is then used to derive special cases in terms of fundamental constant and special functions. The derivations follow the method used by us in [5]. This method involves using a form of the generalized Cauchy's integral formula given by where y, w ∈ C and C is in general an open contour in the complex plane where the bilinear concomitant [5] has the same value at the end points of the contour. The contour C is a counterclockwise oriented simple closed contour around the origin in the complex w-plane. This method involves using a form of Equation (2) then multiplies both sides by a function, then takes the definite integral of both sides. This yields a definite integral in terms of a where Re(a) > 0. The incomplete gamma function has a recurrence relation given by where a = 0, −1, −2, . . . The incomplete gamma function is continued analytically by γ(a, ze 2mπi ) = e 2πmia γ(a, z), where m ∈ Z. When z = 0, Γ(s, z) is an entire function of s and γ(s, z) is meromorphic with simple poles at s = −n for n = 0, 1, 2, . . . with residue (−1) n n! . These definitions are listed in Section 8.2(i) and (ii) in [4].

Hurwitz-Lerch Zeta Function in Terms of the Contour Integral
We use the method in [5]. The cut and contour are in the first quadrant of the complex w-plane with 0 < Re(w + m) < 1. The cut approaches the origin from the interior of the first quadrant and goes to infinity vertically and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy's integral formula we first replace y → log(a) + 1 2 π(2y + 1) then multiply both sides by 1 2 π(−1) y e 1 2 πm(2y+1) and take the infinite sum over y ∈ [0, ∞)to get from Equation (1.232.2) in [8].

The Hurwitz-Lerch Zeta Function in Terms of the Infinite Sum of the Incomplete Gamma Function
Theorem 1. For all k, a, m ∈ C, Proof. Observe the right-hand sides of Equations (5) and (8) are equal so we can equate the left-hand sides and simplify the gamma function to get the stated result.
Proof. Use Equation (9) and set a = 1 and simplify.
Proof. Use Equation (23) and take the first partial derivative with respect to k and set k = 1, a = e π and simplify.
Proof. Use Equation (23) and take the first partial derivative with respect to k and set k = 0, a = e π/2 and simplify.
Proof. Use Equation (9) and take the first partial derivative with respect to k and set m = 0, k = 1, a = e π/2 and simplify.