Elliott's identity and hypergeometric functions

: Balasubramanian R., Naik S., Ponnusamy S., Vuorinen M.

: 2002

: Journal of Mathematical Analysis and Applications

: 271

: 1

: 232

: 256

: 25

: 0022-247X

DOI: https://doi.org/10.1016/S0022-247X(02)00126-9

: http://api.elsevier.com/content/abstract/scopus_id:0036660231

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. © 2002 Elsevier Science (USA). All rights reserved.