Recursion Formula: Given the following formula, a gamma function at one point can be evaluated recursively in terms of its value at another point:
Generalized Factorial: When x is a positive integer, one can easily prove the following by using the recursive formula:
In fact, many formulas involving n! can be extended to non-integer cases by replacing n! with , and that is why the gamma function is also commonly referred to as the generalized factorial function.