A new integral solution of the hypergeometric equation

In this work we derive a new integral of the hypergeometric differential equation, valid for arbitrary parameters α, β, γ and which is expressed in terms of indefinite integrals. This leads to new types of relations between F (α, β, γ;x) and its contiguous functions : F (α, β, γ;x) can be related to only one of the functions contiguous to it.These equations enable one to extend the formulae giving linear, quadratic, cubic,... transformations. A continued fraction due to Gauss for quotients of certain associated hypergeometric series can be related directly to F (α, β, γ;x). A new integral of the confluent hypergeometric equation expressed in terms of indefinite integrals is derived for arbitrary parameters α, γ. This also leads to new types of relations between contiguous confluent hypergeometric functions, similar to those obtained for the hypergeometric function .