Analytical and differential - algebraic properties of Gamma function

Abstract

In this paper we consider some analytical relations between gamma function $\Gamma(z)$ and related functions such as the Kurepa's function $K(z)$ and alternating Kurepa's function $A(z)$. It is well-known in the physics that the Casimir energy is defined by the principal part of the Riemann function $\zeta(z)$ (Blau, Visser, Wipf; Elizalde). Analogously, we consider the principal parts for functions $\Gamma(z)$, $K(z)$, $A(z)$ and we also define and consider the principal part for arbitrary meromorphic functions. Next, in this paper we consider some differential-algebraic $($d.a.$)$ properties of functions $\Gamma(z)$, $\zeta(z)$, $K(z)$, $A(z)$. As it is well-known (H\" older; Ostrowski) $\Gamma(z)$ is not a solution of any d.a. equation. It appears that this property of $\Gamma(z)$ is universal. Namely, a large class of solutions of functional differential equations also has that property. Proof of these facts is reduced, by the use of the theory of differential algebraic fields (Ritt; Kaplansky; Kolchin), to the d.a. transcendency of $\Gamma(z)$.