Abstract
We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.
Highlights
In this present work, we derive a new expression for the Hurwitz zeta function in terms of the infinite sum of the incomplete gamma function given by: ∞ ∑ (−1)n n−2k−1eiπan (−in)k Γ(k + 1, ianπ ) − e−iπank Γ(k + 1, −ianπ ) n =1iπ k+1 ak+1 + 2k+1 (k + 1)ζ −k, = a +1, k+1 where the variables k, a are general complex numbers
We will derive a new expression for the Hurwitz zeta function expressed in terms of the infinite sum of the incomplete gamma function and evaluate this expression in terms of other special functions and fundamental constants not previously published
We derive a new expression for the Hurwitz zeta function in terms of the infinite sum of the incomplete gamma function given by:
Summary
The Hurwitz zeta function ζ (k, a) was again studied but from the perspective of evaluating this function in terms of the infinite sum of the incomplete gamma function Γ(k, a) section (8) in [2] Some examples of this type of evaluation are presented in the works by Kanemitsu et al [3,4] and Bailey et al [5]. With a closer look one sees these infinite sums are taken over the first parameter of the incomplete gamma function In this present paper, we will derive a new expression for the Hurwitz zeta function expressed in terms of the infinite sum of the incomplete gamma function and evaluate this expression in terms of other special functions and fundamental constants not previously published.
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