Browsing by Author "Abd-Elhameed, W. 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.
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    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.