On q-fractional derivatives of Riemann--Liouville and Caputo type

TL;DR

This paper introduces and analyzes fractional q-derivatives of Riemann-Liouville and Caputo types, exploring their properties, relations, and compositions within the framework of q-calculus.

Contribution

It defines new fractional q-derivatives with parametric limits and studies their properties and interrelations, advancing fractional q-calculus theory.

Findings

Defined fractional q-derivatives of Riemann-Liouville and Caputo types

Established properties and relations between these derivatives

Analyzed compositions of the fractional q-derivatives

Abstract

Based on the fractional $q$-integral with the parametric lower limit of integration, we define fractional $q$-derivative of Riemann-Liouville and Caputo type. The properties are studied separately as well as relations between them. Also, we discuss properties of compositions of these operators.