Elliott's identity and hypergeometric functions (Q1849152)

Elliot's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for the complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliot's identity and which also leads to concavity/convexity properties of certain related functions. The authors also show that Elliot's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results the authors obtain a number of identities associated with the Legendre functions of the first and the second kind, respectively.