Multiple-angle formulas of generalized trigonometric functions with two parameters

TL;DR

This paper develops new multiple-angle formulas for generalized trigonometric functions with two parameters, extending classical identities and applying them to generalize topics related to the lemniscate and classical trigonometry.

Contribution

It introduces the first known multiple-angle formulas for these generalized functions, expanding their theoretical framework.

Findings

New multiple-angle formulas established between two types of generalized trigonometric functions.

Formulas applied to generalize classical trigonometric and lemniscate function topics.

Extension of classical identities to generalized functions with two parameters.

Abstract

Generalized trigonometric functions with two parameters were introduced by Dr\'{a}bek and Man\'{a}sevich to study an inhomogeneous eigenvalue problem of the $p$-Laplacian. Concerning these functions, no multiple-angle formula has been known except for the classical cases and a special case discovered by Edmunds, Gurka and Lang, not to mention addition theorems. In this paper, we will present new multiple-angle formulas which are established between two kinds of the generalized trigonometric functions, and apply the formulas to generalize classical topics related to the trigonometric functions and the lemniscate function.