The Non-Newtonian Differential Operators Associated With Iterated Generators

This paper investigates non-Newtonian differentiation associated with integer iterates of generator functions. For m,n∈Z, we consider the derivative operators Dαm,βn and derive explicit first-order representation, transformation, and factorization formulas that relate operators corresponding to different iteration levels. The construction is then extended to higher-order derivatives, where the resulting operators are represented through conjugation with the classical derivative operator Dr. Faà di Bruno’s formula and exponential partial Bell polynomials are used to obtain explicit higher-order expressions. Several examples involving exponential, double-exponential, and quadratic generators illustrate how the general results specialize to familiar non-Newtonian calculi. The results provide a unified treatment of non-Newtonian differentiation based on iterated generators and clarify its relationship with classical differentiation.