Symmetric duality for left and right Riemann-Liouville and Caputo   fractional differences
  A discrete version of the symmetric duality of Caputo-Torres, to relate left and right Riemann-Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional differences, one has to mix between nabla and delta operators. As an application, we derive right fractional summation by parts formulas and left fractional difference Euler-Lagrange equations for discrete fractional variational problems whose Lagrangians depend on right fractional differences. 
Thabet Abdeljawad, Delfim F. M. Torres
math.CA