# Theory of Generalized Trigonometric Functions: from Laguerre to Airy Forms

## TL;DR

This paper introduces a new framework for generalized trigonometric functions using umbral methods, extending classical functions and developing related integral transforms within the context of generalized Borel transforms.

## Contribution

It presents a novel approach to defining generalized trigonometric functions via umbral calculus, expanding the theoretical foundation and introducing new integral transforms.

## Key findings

- New families of generalized trigonometric functions are constructed.
- Extended integral transforms related to these functions are developed.
- The framework connects to generalized Borel transforms.

## Abstract

We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired to the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transform allowing the framing of the previous concepts within the context of generalized Borel tranforms.