“cases? for φ also” a question raised by Ramanujan

Abstract

On pages 51–53 of his lost notebook, S. Ramanujan has recorded several incomplete elliptic integral identities of the first kind. In the process, he raises a question “cases? for [Formula: see text] also”, where he intends to obtain identities involving the [Formula: see text] function. Motivated by this, we obtain two new identities involving the [Formula: see text] function.