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In this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre–Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations. We compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach, through two numerical examples. [ABSTRACT FROM AUTHOR]

Two new formulae expressing explicitly the integrals of Laguerre (Hermite) polynomials of any degree and for any order in terms of the Laguerre (Hermite) polynomials themselves are proved. Another two new explicit formulae relating the Laguerre (Hermite) coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function are also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed. [ABSTRACT FROM AUTHOR]

A formula expressing explicitly the difference derivatives of Hahn polynomials of any degree and for any order in terms of Hahn polynomials themselves is proved. Another explicit formula, which expresses the Hahn expansion coefficients of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its original Hahn coefficients, is also given. A formula for the Hahn coefficients of the moments of one single Hahn polynomial of certain degree is proved. A formula for the Hahn coefficients of the moments of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its Hahn coefficients is also obtained. Application of these formulae for solving ordinary difference equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Hahn-Hahn, Meixner-Hahn, Kravchuk-Hahn and Charlier-Hahn is also developed. [ABSTRACT FROM AUTHOR]

Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials { P n ( x ): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n ( x ) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f ( x ), and for the moments x ℓ ∇ q f ( x ), of an arbitrary function f ( x ) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica), in order to build and solve recursively for the connection coefficients between two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed. [ABSTRACT FROM AUTHOR]

A formula expressing the Hermite coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Hermite polynomials of any degree and for any order as a linear combination of suitable Hermite polynomials is deduced. A formula for the Hermite coefficients of the moments of one single Hermite polynomial of certain degree is given. Formulae for the Hermite coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Hermite coefficients are also obtained. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in Hermite coefficients, are discussed. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi-Hermite and Laguerre-Hermite polynomials is described. Explicit formula for these coefficients between Jacobi and Hermite polynomials is given, of which the ultraspherical polynomials and the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Analytical formula for the connection coefficients between Laguerre and Hermite polynomials is also obtained. [ABSTRACT FROM AUTHOR]