The generalized gamma functions

Abstract

In this paper, we introduce a way to generalize the Euler's gamma function as well as some related special functions. With a given polynomial in one variable $f(t)\ge 0$, we can associate a function, so-called "gamma function associated with $f$", defined by ${\Gamma }_f(s):={\int }_0^{\infty }{f}^{s-1}{e}^{-t}dt$. This function has many features similar to the Euler's gamma function. We also present some initial results on the gamma-type functional equation for ${\Gamma }_f(s)$ in some special cases.

13 pages, accepted for publication in AMV