On a Result Contiguous to Ramanujan’s Entry 35 (iii)

Abstract

One of the entries in Ramanujan's second notebook [2, entry 35 (iii), p. 99], for arbitrary n:

(1-x2)*(-1/2) cos (2n sin *(-1) x)=2F1(1/2+n,1/2-n;1/2;x2).

The aim of this short research note is to provide the following result closely related to Ramanujan's entry viz, for arbitrary n:

(1-x2)*(-1/2) sin(2n sin*(-1) x)=2nx 2F1(1+n,1-n;3/2;x2)

The result is derived with the help of classical Saalschutz's summation theorem for the series 3F2 and may be potentially useful. In addition to this, we also provide a very elementary proofs of (a) and (b).