"Timespace" --

There is no such thing as a "fractional factorial". The Gamma function for non-negative real arguments is a smooth and continuous function that passes through all the discrete points of the factorial function, which is defined only for non-negative 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*(n-1)! -- or alternatively, (n+1)! = (n+1)*n!. Defining 0! = 1 provides the basis for making the definition functional, and equivalent to n! = n*(n-1)*(n-2)*...*1.

Gamma is combined with factorial as *x!* on menuless or limited-menu models in order to conserve keyboard space.

Factorial (*n!*) is separated from Gamma on the HP-42S because the expansive menus make it feasible.

Note: Lower-end and business models, such as the HP-10C, HP-12C, and HP-17B* -- don't provide Gamma. These have only *n!*.

-- KS

*Edited: 28 Sept 2006, 11:38 p.m. *