On gamma functions with respect to the alternating Hurwitz zeta functions

In 1730, Euler defined the Gamma function Γ(x) by the integral representation. It possesses many interesting properties and has wide applications in various branches of mathematics and sciences. According to Lerch, the Gamma function Γ(x) can also be defined by the derivative of the Hurwitz zeta function ζ(z,x)=∑ₙ₌₀∞1(n+x)ᶻ at $z=0$. Recently, Hu and Kim defined the corresponding Stieltjes constants γₖ(x) and Euler constant γ₀ from the Taylor series of the alternating Hurwitz zeta function ζE(z,x) ζE(z,x)=∑ₙ₌₀∞(-1)ⁿ/(n+x)ᶻ. And they also introduced the corresponding Gamma function Γ(x) which has the following Weierstrass--Hadamard type product Γ(x)=1/xe^γ₀x∏ₖ₌₁∞(e-x/k(1+x/k))^(-1)ᵏ⁺¹. In this paper, we shall further investigate the function Γ(x), that is, we obtain several properties in analogy to the classical Gamma function Γ(x), including the integral representation, the limit representation, the recursive formula, the special values, the log-convexity, the duplication formula and the reflection equation. Furthermore, we also prove a Lerch-type formula, which shows that the derivative of ζE(z,x) can be representative by Γ(x). As an application to Stark's conjecture in algebraic number theory, we will explicit calculate the derivatives of the partial zeta functions for the maximal real subfield of cyclotomic fields at $z=0$.