Mathematics – General Mathematics
Scientific paper
2006-05-16
International Journal of Applied Mathematics & Statistics, Vol.11, No. 7, November 2007, 118-129
Mathematics
General Mathematics
Paper by invitation for The Special Volume dedicated to the Tricentennial Birthday Anniversary of L. Euler, 2007
Scientific paper
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 $\Gamma(z)$.
Malesevic Branko
Mijajlović Žarko
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