nt.number theory - Why is the Gamma function shifted from the factorial by 1?

I would also go for $Pi(t)$ or $t!$, but a possible reason to prefer the shifted version, $Gamma(t)$, is the following. The gamma densities $gamma_t$, $tin mathbb{R}$ defined as

$$gamma_t(x):=frac{x_+^{t-1}e^{-x}}{Gamma(t)},$$

are a convolution semigroup, so that $Gamma(t)$ appears naturally as the normalization factor of $gamma_t$. (And, of course, the semigroup relation

$$gamma_t*gamma_s=gamma_{t+s}$$

would be destroyed shifting from $t-1$ to $t$ in the definition of $gamma_t$)

Also note that the expression of the Beta function

$$B(t,s):=int_0^1 x^{t-1}(1-x)^{s-1} , dx$$

in terms of the $Gamma$ function, if shifted, would also loose the useful form

$$B(t,s)=frac{Gamma(t)Gamma(s)}{Gamma(t+s)}.$$

(incidentally note that this relation follows plainly from the semigroup property since as a general fact, the integral of a convolution of two functions is the product of their integrals).