Actually there are other (less-frequently) used extensions to the factorial, with different properties from the gamma function which may be desirable in some contexts.

Euler's Gamma Function http://www.luschny.de/math/factorial/hadamard/f1.gif
Euler's Gamma Function

Hadamard's Gamma function http://www.luschny.de/math/factorial/hadamard/hadgamma1.gif
Hadamard's Gamma function

Luschny's factorial function http://www.luschny.de/math/factorial/hadamard/LuschnyFactorial5.gif
Luschny's factorial function

See here for more information.

Actually there are other (less-frequently) used extensions to the factorial, with different properties from the gamma function which may be desirable in some contexts.

Euler's Gamma Function

Hadamard's Gamma function

Luschny's factorial function

See here for more information.