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We give a differential calculus defined on deformed generalized Fibonaccipolynomials. The main goal is to generalize the $q$-calculus and the Goldencalculus or Fibonacci calculus and thus obtain the Pell calculus, Jacobsthalcalculus, Chebysheff calculus, Mersenne calculus, among others. This calculuswill serve as a framework for the solutions of equations in differences withproportional delay. For this reason, we define the deformed $(s,t)$-exponentialfunctions and we also construct a family of functions that are solutions of alinear functional difference equation with proportional delay of first order.

Differential Calculus on Deformed Generalized Fibonacci Polynomials and the Functional-Difference Equation $\mathbf{D}_{s,t}f(x)=af(ux)$