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 $f(t) \geq 0$, 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.