A look at generalized trigonometric functions as functions of their two parameters and further new properties

TL;DR

This paper explores the monotonicity and convexity properties of generalized trigonometric and hyperbolic functions with two parameters, introduces new hypergeometric representations, and provides explicit integral evaluations.

Contribution

It fills a gap by analyzing parameter-dependent properties of these functions and presents new hypergeometric representations and integral formulas.

Findings

Established (log)-convexity/concavity conditions for some functions

Derived two new hypergeometric representations for generalized cosine functions

Provided four explicit integral evaluations involving these functions

Abstract

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions.