### Abstract

Generalized trigonometric functions are applied to Legendre’s 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 p in terms of the arithmeticgeometric mean, in the classical way as the Gauss-Legendre algorithm for p by Brent and Salamin. Moreover, an elementary alternative proof of Ramanujan’s cubic transformation is also given.

Original language | English |
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Pages (from-to) | 202-226 |

Number of pages | 25 |

Journal | Kodai Mathematical Journal |

Volume | 39 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Mar 25 |

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### Keywords

- Arithmetic-geometric mean
- Gauss-Legendre’s algorithm
- Generalized complete elliptic integrals
- Generalized trigonometric functions
- P-Laplacian
- Ramanujan’s cubic transformation

### ASJC Scopus subject areas

- Mathematics(all)