Discrete fractional Calculus and Inequalities

TL;DR

This paper introduces a new Caputo-like discrete fractional difference, compares it with existing Riemann-Liouville analogs, develops fractional Taylor formulas, and derives related inequalities, advancing discrete fractional calculus theory.

Contribution

It is the first to produce discrete fractional Taylor formulas and estimate their remainders, expanding the tools available in discrete fractional calculus.

Findings

Defined a Caputo-like discrete fractional difference

Compared it with Riemann-Liouville fractional difference

Derived discrete fractional Taylor formulas and inequalities

Abstract

Here we define a Caputo like discrete fractional difference and we compare it to the earlier defined Riemann-Liouville fractional discrete analog. Then we produce discrete fractional Taylor formulae for the first time, and we estimate their remainders. Finally, we derive related discrete fractional Ostrowski, Poincare and Sobolev type inequalities.