Elliott's identity and hypergeometric functions

Abstract

Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for 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 Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively.