Some double-angle formulas related to a generalized lemniscate function

TL;DR

This paper derives double-angle formulas for a generalized lemniscate function, which is the inverse of an integral related to Ramanujan's work, expanding the mathematical understanding of such special functions.

Contribution

It introduces new double-angle formulas for a generalized lemniscate function, connecting classical integral functions with Ramanujan's notebooks.

Findings

Derived double-angle formulas for the generalized lemniscate function

Connected the function to Ramanujan's integral representations

Enhanced understanding of inverse functions related to elliptic integrals

Abstract

In this paper we will establish some double-angle formulas related to the inverse function of $\int_0^x dt/\sqrt{1-t^6}$. This function appears in Ramanujan's Notebooks and is regarded as a generalized version of the lemniscate function.