The Gaussian Hypergeometric Differential Equation is:
where , , and are constants. The indicial equation of the hypergeometric differential equation is:
which has the roots and . Using the Frobenius method, the series solution for can be express as:
where and the series converges for . This series is called Hypergeometric Series. The sum of the hypergeometric series denoted by is called Hypergeometric Function, which is:
General Solution: If , , and are all non-integers, the general solution for the hypergeometric differential equation is:
which is valid for .
Gamma Function: A hypergeometric function can be expressed in terms of gamma functions.