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 $\gamma_t$, $t\in \mathbb{R}$ 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).

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 $\gamma_t$, $t\in \mathbb{R}$ 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).

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 $\gamma_t$, $t\in \mathbb{R}$ 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, it you feel like tryin the short computation).

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 $\gamma_t$, $t\in \mathbb{R}$ 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).