Browsing by Author "Ahmed, H. M."

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THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS OF DOUBLE AND TRIPLE JACOBI POLYNOMIALS
(SPRINGER SINGAPORE PTE LTD, 2012) Doha, E. H.; Abd-Elhameed, W. M.; Ahmed, H. M.
Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspherical polynomials are considered. Also the results for Chebyshev polynomials of the first, second, third and fourth kinds and of Legendre polynomials are noted. An application of how to use double Jacobi polynomials for solving Poisson's equation in two variables subject to nonhomogeneous mixed boundary conditions is described.
Derivation of the errors involved in Hermite interpolation and their applications to quadrature formulae
(ELSEVIER SCIENCE INC, 2006) Ashour, S. A.; Ahmed, H. M.
Divided differences are presented to derive closed-form for the error involved in Hermite interpolation formulae for the class of functions, which need not be sufficiently differentiable. As applications of these formulae the error estimates in the known Gaussian quadrature formulae for regular and singular integrals are taken up. Some examples are considered for numerical experiments
Explicit formulae for the coefficients of integrated expansions of Laguerre and Hermite polynomials and their integrals
(TAYLOR & FRANCIS LTD, 2009) Doha, E. H.; Ahmed, H. M.; El-Soubhy, S. I.
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.
Linearization formulae for certain Jacobi polynomials
(SPRINGER, 2016) Abd-Elhameed, W. M.; Doha, E. H.; Ahmed, H. M.
In this article, some new linearization formulae of products of Jacobi polynomials for certain parameters are derived. These new derived formulae are expressed in terms of hypergeometric functions of unit argument, and they generalize some existing formulae in the literature. With the aid of some standard formulae and also by employing symbolic algebraic computation, and in particular Zeilberger's algorithm, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence several linearization formulae of products of Jacobi polynomials for special parameters free of hypergeometric functions are deduced.
Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials
(TAYLOR & FRANCIS LTD, 2009) Ahmed, H. M.
The two formulae expressing explicitly the difference derivatives and the moments of discrete orthogonal polynomials {Pn(x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of Pn(x) themselves are proved. Two other formulae for the expansion coefficients of general-order difference derivatives qf(x), and for the moments xqf(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 Hahn-Charlier, Hahn-Meixner and Hahn-Kravchuk are described.
Recurrence relation approach for expansion and connection coefficients in series of Hahn polynomials
(TAYLOR & FRANCIS LTD, 2006) Doha, E. H.; Ahmed, H. M.
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.