# Applications of generalized trigonometric functions with two parameters II

## TL;DR

This paper explores the applications of two-parameter generalized trigonometric functions (GTFs) in differential equations, hypergeometric functions, and inequalities, extending their use beyond the single-parameter case.

## Contribution

It introduces new applications of two-parameter GTFs to complex differential equations, hypergeometric formulas, and inequalities, which were previously limited.

## Key findings

- Application of GTFs to primitive equations of oceanic and atmospheric dynamics
- Derivation of new formulas for Gaussian hypergeometric functions
- Establishment of the $L^q$-Lyapunov inequality for the $p$-Laplacian

## 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. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.