Analytical and Differential-algebraic Properties of Gamma Function

In this paper we consider some analytical relations between gamma function Γ(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 ζ(z) (Blau, Visser, Wipf; Elizalde). Analogously, we consider the principal parts for functions Γ(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 Γ(z), ζ(z), K(z), A(z). As it is well-known (Hölder; Ostrowski) Γ(z) is not a solution of any d.a. equation. It appears that this property of Γ(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 Γ(z). 1 Analytical properties In this section we consider analytical properties of the gamma and related functions which pertain to the principal part of a function at a point. 1.1 The principal part of the gamma function Gamma function is defined by the integral: