Hypergeometric Functions

Hypergeometric function is a special function defined by the hypergeometric series. It is the solution of a linear second-order ordinary differential equation called the hypergeometric equation. Majority of the second-order ordinary linear differential equations encountered in science and engineering can be expressed in terms of the three parameters (a, b, c) of the hypergeometric equation and its transformations. Applications of the four Euler transformations to the six Kummer solutions give all the possible 24 forms of the solutions of the hypergeometric equation. Majority of the special functions can be represented in terms of hypergeometric functions. The Legendre polynomials can be expressed in terms of the hypergeometric functions. Pochhammer symbols are very useful in manipulations with hypergeometric functions. Pochhammer notation also is very useful in getting rid of a parameter in the numerator or the denominator of the hypergeometric function.