Studies in fractal–fractional operators with examples

Abstract

By using the generalization of the gamma function ($p$-gamma function: ${\Gamma }_p(.)$), we introduce a generalization of the fractal–fractional calculus which is called $p$-fractal fractional calculus. We extend the proposed operators into the symmetric complex domain, specifically the open unit disk. Normalization for each operator is formulated. This allows us to explore the most important geometric properties. Examples are illustrated including the basic power functions.