Fractional \(h\)-difference equations arising from the calculus of variations (Q2853198)

The authors deal with the definitions and properties of the left and right fractional \(h\)-differences. Left and right fractional \(h\)-differences represent the discrete versions of the Riemann-Liouville left and right fractional derivatives, while the fractional \(h\)-sum represents the discrete version the fractional integral. The properties of the left and right fractional \(h\)-differences prove to be analogous to the properties of the left and right Riemann-Liouville derivatives. The central role plays the fractional \(h\)-difference of the power-type function, as well as the exponent law. The authors also obtain the function whose left (right) fractional \(h\)-difference is zero. These results are used in order to formally solve Euler-Lagrange equations in two simple cases of given Lagrangians.