The generalized gamma functions

TL;DR

This paper introduces a generalized gamma function associated with a polynomial, extending Euler's gamma function, and explores its properties and functional equations.

Contribution

It proposes a new class of gamma functions linked to polynomials and investigates their fundamental properties and functional equations.

Findings

Defined gamma functions associated with polynomials

Established initial properties similar to Euler's gamma function

Explored functional equations in special cases

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.