A new form of the generalized complete elliptic integrals

TL;DR

This paper introduces a new form of generalized complete elliptic integrals using generalized trigonometric functions, enabling easier computation of generalized pi and providing a new proof of Ramanujan's cubic transformation.

Contribution

It presents a novel form of generalized elliptic integrals and demonstrates their properties, including a computation formula for generalized pi and a new proof of Ramanujan's cubic transformation.

Findings

New form of generalized elliptic integrals introduced

Computation formula for generalized pi established

Elementary proof of Ramanujan's cubic transformation provided

Abstract

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $\pi$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $\pi$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.