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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:

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

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?

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Joseph O'Rourke
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    http://mathoverflow.net/questions/20960/why-is-the-gamma-function-shifted-from-the-factorial-by-1/20962#20962 – Will Jagy Feb 02 '14 at 22:42
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    Because when Legendre wrote a letter to Gauss about the Legendre $L$ function, Gauss happened to have mistakenly viewed it in an upside down mirror.... sorry, jk! – Suvrit Feb 03 '14 at 00:23
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    Not the historical explanation, but merely an (interesting ?) observation: Gamma is the Greek letter for 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. – Lucian Feb 03 '14 at 03:40

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The first use of $\Gamma$ in this sense is due to Legendre (1811). It is unknown why he choose that notation but some speculations are recorded at http://jeff560.tripod.com/functions.html

They range from the Gamma being an inverted L (from Legendre) and variants, to the in my opinion more integresting that the logarithm of $\Gamma$ is related to the Euler-Mascheroni constant, which is mentioned in that work of Legendre and was (then) denoted $C$ (which makes $\Gamma$ as a Greek capital C natural) or also $\gamma$ (that usage was present before the work of Legendre too).