Re: Factorials on the Free42 Message #9 Posted by Karl Schneider on 27 Sept 2006, 9:50 p.m., in response to message #4 by Timespace
"Timespace" 
There is no such thing as a "fractional factorial". The Gamma function for nonnegative real arguments is a smooth and continuous function that passes through all the discrete points of the factorial function, which is defined only for nonnegative integers.
By definition, Gamma(x+1) = Integral (0, infinity, t^{x}e^{t}dt), so
Gamma(1) = 0! = 1
Gamma(2) = 1! = 1
Gamma(3) = 2! = 2
Gamma(4) = 3! = 6
and so forth...
I'm not certain what the basis of the definitions are, but I surmise that Gamma(x) is defined such that its first discontinuity as x decreases is at x = 0, instead of at x = 1. This effectively separates the behavior of the gamma function into regions of "positive x" and "negative x".
Factorials can be defined recursively such that n! = n*(n1)!  or alternatively, (n+1)! = (n+1)*n!. Defining 0! = 1 provides the basis for making the definition functional, and equivalent to n! = n*(n1)*(n2)*...*1.
Gamma is combined with factorial as x! on menuless or limitedmenu models in order to conserve keyboard space.
Factorial (n!) is separated from Gamma on the HP42S because the expansive menus make it feasible.
Note: Lowerend and business models, such as the HP10C, HP12C, and HP17B*  don't provide Gamma. These have only n!.
 KS
Edited: 28 Sept 2006, 11:38 p.m.
