Analytical and differential - algebraic properties of Gamma function

Mijajlovic, Zarko; Malesevic, Branko

Analytical and differential - algebraic properties of Gamma function

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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 (H\" older; 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 $Γ(z)$.
Paper by invitation for The Special Volume dedicated to the Tricentennial Birthday Anniversary of L. Euler, 2007