Fractional q-Calculus

Fractional -calculus extends fractional calculus and q-calculus by defining noninteger-order versions of the q-derivative and q-integral. It includes -analogues of the Riemann-Liouville fractional derivative and Caputo fractional derivative and the corresponding fractional -difference equations.

As in ordinary fractional calculus, several conventions are used, including left- and right-sided operators, forms based on the Riemann-Liouville fractional derivative or Caputo fractional derivative, and delta and nabla operators on geometric -time scales. The resulting kernels are expressed using the q-gamma function and q-factorials (Atici and Eloe 2007, Annaby and Mansour 2012).