Browsing by Author "Ahmed, HM"

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Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of Al-Salam-Carlitz I polynomials
(IOP PUBLISHING LTD, 2005) Doha, EH; Ahmed, HM
Two formulae expressing explicitly the derivatives and moments of Al-Salam-Carlitz I polynomials of any degree and for any order in terms of Al-Salam-Carlitz I themselves are proved. Two other formulae for the expansion coefficients of general-order derivatives D-q(p) f(x), and for the moments x(e)D(q)(p) f (x), of an arbitrary function f (x) in terms of its original expansion coefficients are also obtained. Application of these formulae for solving q-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 Al-Salam-Carlitz I polynomials and any system of basic hypergeometric orthogonal polynomials, belonging to the q-Hahn class, is described.
On the coefficients of integrated expansions of Bessel polynomials
(ELSEVIER SCIENCE BV, 2006) Doha, EH; Ahmed, HM
A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed. (c) 2005 Elsevier B.V. All rights reserved.
Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials
(IOP PUBLISHING LTD, 2004) Doha, EH; Ahmed, HM
A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential 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 Bessel-Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Bessel and Hermite-Bessel are also developed.
Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials
(TAYLOR & FRANCIS LTD, 2006) Doha, EH; Ahmed, HM
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 del(q) f(x), and for the moments x(l)del(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.