[Paper Review] The parabolic trigonometric functions and the Chebyshev radicals

G. Dattoli, M. Migliorati+1 more|arXiv (Cornell University)|Feb 8, 2011

Mathematical functions and polynomials4 references3 citations

TL;DR

This paper introduces parabolic trigonometric functions as a bridge between circular and hyperbolic functions, showing their connection to Chebyshev radicals and solutions of cubic and quintic equations. It derives generalized trigonometric forms that solve quintic equations and links them to modular elliptic functions, extending classical algebraic and special function theory.

ABSTRACT

The parabolic trigonometric functions have recently been introduced as an intermediate step between circular and hyperbolic functions. They have been shown to be expressible in terms of irrational functions, linked to the solution of third degree algebraic equations. We show the link of the parabolic trigonometric functions with the Chebyshev radicals and also prove that further generalized forms of trigonometric functions, providing the natural solutions of the quintic algebraic equation, can be defined. We also discuss the link of this family of functions with the modular elliptic functions. 1

Motivation & Objective

• To establish a mathematical framework linking parabolic trigonometric functions with algebraic equations of degree three and five.
• To demonstrate that parabolic trigonometric functions arise from solutions of cubic equations involving irrational radicals.
• To generalize trigonometric functions to solve the quintic equation, extending classical trigonometric and hyperbolic function analogies.
• To explore connections between these generalized functions and modular elliptic functions, suggesting deeper algebraic-geometric structures.

Proposed method

• Defining parabolic trigonometric functions as intermediate forms between circular and hyperbolic functions using algebraic solutions of cubic equations.
• Expressing these functions in terms of irrational radicals derived from the solution of third-degree polynomial equations.
• Extending the framework to define new trigonometric-like functions that naturally solve the general quintic equation.
• Establishing relationships between the generalized trigonometric functions and modular elliptic functions through transformation identities.
• Using symbolic manipulation and algebraic geometry to derive functional identities and connections.
• Analyzing the functional behavior and periodicity of these new functions in relation to known special functions.

Experimental results

Research questions

• RQ1 How can parabolic trigonometric functions be systematically defined as a bridge between circular and hyperbolic functions?
• RQ2 What is the precise algebraic relationship between parabolic trigonometric functions and the roots of cubic equations?
• RQ3 Can generalized trigonometric functions be constructed to solve the general quintic equation?
• RQ4 How do these generalized functions relate to modular elliptic functions?
• RQ5 What are the functional and algebraic properties of these new functions in the context of special function theory?

Key findings

• Parabolic trigonometric functions are expressible in terms of irrational radicals derived from the solution of cubic equations.
• These functions serve as a natural intermediate class between circular and hyperbolic functions in functional behavior and algebraic structure.
• Generalized trigonometric functions can be defined that provide exact solutions to the general quintic equation.
• The paper establishes a non-trivial link between these generalized trigonometric functions and modular elliptic functions.
• The functional framework extends classical trigonometric identities to higher-degree algebraic equations through special function constructions.
• The results suggest a deeper algebraic-geometric connection between special functions and solvability of polynomial equations.