Some implications of a new definition of the exponential function on time scales

TL;DR

This paper introduces a new exponential function on time scales, explores its properties, and demonstrates its applications in numerical schemes, q-calculus, and Lie group systems, advancing the mathematical framework for dynamic equations.

Contribution

It presents the Cayley-exponential function on time scales, linking it to numerical methods and extending to Padé approximants, offering a unified approach to exponential functions in dynamic systems.

Findings

Cayley-exponential relates to implicit midpoint and trapezoidal rules.

Extension to Padé-exponential functions provides a broader class of solutions.

Exact exponential function on time scales is defined and applied.

Abstract

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Pad\'e approximants, we obtain Pad\'e-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.