Bastos, N. R. O.Ferreira, R. A. C.Torres, D. F. M.2014-12-092014-12-0920110165-1684http://hdl.handle.net/10400.19/2433We introduce a discrete-time fractional calculus of variations on the time scale (hℤ)a,a∈ℝ,h>0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when h tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation. © 2010 Elsevier B.V. All rights reserved.engCalculus of variationsEuler-Lagrange equationFractional difference calculusFractional summation by partsLegendre necessary conditionNatural boundary conditionsTime scale hZjournal article