《Integral Transforms and Special Functions》2012,23(4):349-361

The main purpose of this paper is to present a family of finite summation formulas and to apply it in order to derive several functional relationships involving various multivariable hypergeometric polynomials and the Gauss hypergeometric function. A number of special and limit cases of these functional relationships are also considered.  相似文献   

    

《Integral Transforms and Special Functions》2012,23(1):13-29

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.  相似文献   

    

《Integral Transforms and Special Functions》2012,23(10):769-773

Additive formulae of Jacobin theta functions are obtained. As applications, we derive the Ramanujan’s modular equations of degree three and five.  相似文献   

    

《Integral Transforms and Special Functions》2012,23(1):70-83

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《Integral Transforms and Special Functions》2012,23(6):411-425

A two-variable generalization of the Big -1 Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big -1 Jacobi polynomials. Their orthogonality measure is obtained. Their bispectral properties (eigenvalue equations and recurrence relations) are determined through a limiting process from the two-variable Big q-Jacobi polynomials of Lewanowicz and Woźny. An alternative derivation of the weight function using Pearson-type equations is presented.  相似文献   

    

《Integral Transforms and Special Functions》2012,23(4):253-266

The paper's main result is a simple derivation rule for the Jacobi polynomials with respect to their parameters, i.e. for [image omitted] . It is obtained via relations for the Guassian hypergeometric function concernuing parameter derivatives and integer shifts in the first two arguments. These are of interest on their own for further applications to continuous and discrete orthogonal polynomials. The study is motivated by a Galerkin method with moving weight, presents all proffs in detail, and terminated in a brief discussion of the generated polynomials.  相似文献   

    

《Integral Transforms and Special Functions》2012,23(1-2):87-96

Expansions of continuous and discrete Bernsein bases on shifted Jacobi and Hahn polynomials, respectively, are explicitly obtained in terms of Hahn-Eberlein orthogonal polynomials. The basic tool is an algorighm, recently developed by the authors, which allows one to solve the connection problem between two families of polynomials recurrently. ^{∗ ∗}This work was partially supported by NATO grant CRG-960213. The work of I.A. and E. G. has been partially finantial supported by Xunta de Galicia-Universidade de Vigo under grant 64502I703. A.Z. also wish to acknowledge partial finaltial support from DGES (Ministerio de Educación y Cultura of Spain) under contract PG95-1205.   相似文献   

    

《Journal of Pure and Applied Algebra》2024,228(2):107487

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Umidjon Durdiev  Zhanna Totieva 《Mathematical Methods in the Applied Sciences》2019,42(18):7440-7451

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.  相似文献   

    

《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(5):1361-1399

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.  相似文献