I am aware of some of the history of the gamma function $\Gamma(z)$, partly through a 2009(!) MO question "Who invented the gamma function?"—Euler, Bernoulli, etc. My question does not seem to be answered in that discussion, or in other historical accountings I can easily locate:

. Why was the symbol $\Gamma$ chosen for the generalized factorial?Q

Were $\alpha(z)$ and $\beta(z)$ already "taken" and so $\Gamma$ was a natural successor? Or was the choice due to the shape of the uppercase $\Gamma$? Or some other reason? Or lost to history?

G. The Gaussian function is $\int_0^\infty e^{-x^2}dx=\Gamma\big(1+\frac12\big)$. Generalizing, we have $\int_0^\infty e^{-x^n}dx=\Gamma\big(1+\frac1n\big)$, as can be easily proven by a simple variable change. So, whatever the actual reason for its name, I find its mathematical and symbolical connection with the Gaussian integral providential. On a related note, I also like to see a symbolical link between the name of the Beta function and Binomial coefficients. $\endgroup$