Lucas-Pantograph Type Exponential, Trigonometric, and Hyperbolic Functions
Ronald Orozco López
TL;DR
This work extends Pantograph-type deformations to Lucas sequences, defining $exp_{s,t}(x,u)$ as a Lucas-analogue of the Pantograph exponential and deriving Euler-type relations. It builds a comprehensive Lucas-calculus framework including the Lucas-derivative $\mathbf{D}_{s,t}$, Lucas-Integral, and $(u,v)$-deformed Lucasnomials and multinomials. The authors develop a complete suite of Lucas-Pantograph special functions: exponential, trigonometric, and hyperbolic, with parity, addition/subtraction, Pythagorean, double-angle, special-value, and periodicity identities. The framework links deformations of classical functions to families like Fibonacci, Pell, Jacobsthal, and Chebyshev polynomials, offering potential applications across combinatorics, number theory, and statistical mechanics.
Abstract
In this paper, we include some new results for the Lucas calculus. A Lucas-Pantograph type exponential function is introduced. Additionally, we define Lucas-Pantograph type trigonometric functions, and some of their most notable identities are given: parity, sum and difference formulas, Pythagorean identities, double-angle identities, and some special values. Lucas-Pantograph type hyperbolic functions are also introduced.