Abstract
This chapter discusses existence theorems for linear differential equations and presents equations containing a real or complex parameter. Hypergeometric function is a special function represented by the hypergeometric series that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation. Every second-order linear ordinary differential equation with three regular singular points can be transformed into this equation. There is no known system for organizing all of the identities involving the hypergeometric function; also, there is no known algorithm that can generate all identities. However, a number of different algorithms are known that generate different series of identities. The Legendre differential equation is a second-order ordinary differential equation and has two linearly independent solutions.
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