Existence of Solutions of Functional-Difference Equations with Proportional Delay on Deformed Generalized Fibonacci Polynomials via Successive Approximation and Bell Polynomials

In this paper, we study the existence of solutions of the functional difference equations with proportional delay on deformed generalized Fibonacci polynomials via successive approximation method and Bell polynomials. First, we introduce the deformed generalized Fibonacci polynomials and show that the $q$-numbers can be viewed as "bifurcation" of deformed $(s,t)$-numbers. These deformations are closely related to proportional delay. Second, a differential and integral calculus on deformed generalized Fibonacci polynomials is introduced. The main reason for introducing this calculation is to have a framework for solving proportional functional equations and thus obtain the Pell calculus, Jacobsthal calculus, Chebysheff calculus, and Mersenne calculus, among others. We study the convergence of $(s,t)$-exponential type series and its dependence on the deformation parameter. We define the deformed $(s,t)$-exponential functions and we give its analytic and algebraic properties. In addition, we study the $(1,u)$-deformed $(s,t)$-exponential function and use it to prove the existence of functional difference equations with proportional delay. The solution is not unique when it is related to $q$-periodic functions.