Multiplicative Riemann–Liouville fractional integrals and derivatives

Abstract

This study explores the connections between fractional calculus, a field that has recently garnered significant research interest, and multiplicative analysis. The introduction provides a comprehensive overview of the historical development and foundational concepts of these areas. The preliminary section outlines key definitions and illustrative examples from multiplicative analysis. The research derives the multiplicative representations of the gamma and beta functions and examines their fundamental properties. Furthermore, generalizations of integrals and derivatives within the framework of multiplicative analysis are formulated, accompanied by explicit formulas for multiplicative integrals and derivatives. Finally, fractional-order multiplicative integral derivatives for selected functions are introduced and visualized through graphical representations, highlighting their practical implications.