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      A Generalizedq-Mittag-Leffler Function byq-Captuo Fractional Linear Equations

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          Abstract

          Some Caputo q-fractional difference equations are solved. The solutions are expressed by means of a new introduced generalized type of q-Mittag-Leffler functions. The method of successive approximation is used to obtain the solutions. The obtained q-version of Mittag-Leffler function is thought as the q-analogue of the one introduced previously by Kilbas and Saigo (1995).

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          Initial value problems in discrete fractional calculus

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            Discrete-Time Fractional Variational Problems

            We introduce a discrete-time fractional calculus of variations on the time scale \(h\mathbb{Z}\), \(h > 0\). 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.
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              Some Fractional q-Integrals and q-Derivatives

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                Author and article information

                Journal
                Abstract and Applied Analysis
                Abstract and Applied Analysis
                Hindawi Limited
                1085-3375
                1687-0409
                2012
                2012
                : 2012
                :
                : 1-11
                10.1155/2012/546062
                © 2012

                http://creativecommons.org/licenses/by/3.0/

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