The theory of complex Dirichlet spaces is very important in complex analysis. These spaces consist of holomorphic functions whose derivatives are Lebesgue square-integrable in norm. Similarly, the theory of quaternionic Dirichlet modules of slice regular functions is widely known as a natural extension of the theory of complex Dirichlet spaces via quaternionic analysis.

On the other hand, fractional calculus is a theory that allows us to consider integrals and derivatives of any real or complex order, where the fundamental theorem of usual calculus is also extended. Then, by extending the concept of derivative through the fractional proportional derivative with respect to the truncated exponential function in the complex Dirichlet spaces and quaternionic Dirichlet modules, we obtain many families of function spaces.

Some families of complex Banach spaces and quaternionic Banach modules of functions associated with generalized fractional derivatives with respect to a truncated exponential function are presented. The aim of this study is to present an interesting extension of the usual function theory of complex Dirichlet spaces and quaternionic Dirichlet modules in terms of fractal– fractional calculus, finding a large variety of families of complex Banach spaces and quaternionic Banach modules that contain the already known spaces.