The complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p=4$

TL;DR

This paper explores the generalization of elliptic integrals using the generalized trigonometric function for p=4, deriving a formula for computing _p based on the arithmetic-geometric mean, extending classical  formulas.

Contribution

It provides a novel computation formula for _p when p=4, linking generalized elliptic integrals with the arithmetic-geometric mean, a connection not previously established.

Findings

Derived a _p computation formula for p=4

Connected generalized elliptic integrals with the arithmetic-geometric mean

Extended classical  formulas to the p=4 case

Abstract

The complete $p$-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function $\sin_p{\theta}$ and its half-period $\pi_p$. It is shown, only for $p=4$, that the generalized $p$-elliptic integrals yield a computation formula of $\pi_p$ in terms of the arithmetic-geometric mean. This is a $\pi_p$-version of the celebrated formula of $\pi$, independently proved by Salamin and Brent in 1976.