# Applications of generalized trigonometric functions with two parameters

## TL;DR

This paper explores the use of generalized trigonometric functions with two parameters in solving nonlinear nonlocal boundary value problems, extending their applications beyond the $p$-Laplacian, and derives new integral formulas related to lemniscate functions.

## Contribution

It introduces novel applications of two-parameter GTFs to differential equations unrelated to the $p$-Laplacian and provides new integral formulas including Wallis-type formulas.

## Key findings

- Application of GTFs to non-$p$-Laplacian problems
- Derivation of Wallis-type integral formulas
- Connections to lemniscate functions and constants

## Abstract

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the $p$-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without $p$-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.