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Etude théorique et algorithmique des séries de Chebyshev solutions d'équations différentielles holonomes

Abstract : The first part of this thesis deals with the manipulation of orthogonal series with computer algebra. Using the hypergeometrical approach, we obtain in a constructive and synthetic manner difference operators which define elementary operations on orthogonal series such as multiplication by a polynomial, differentiation ou evaluation of truncated series. These elementary operations have been implemented in Maple as primitives on which more complicated operations are built : application of a differential operator, products of series and particularly solution of differential problems by means of tau-methods. In the case of Chebyshev series, the results of the first part permit us to build a difference equation, a so-called Chebyshev recurrence, verified by Chebyshev coefficients of any function which satisfies a given holonomic differential equation. Serveral problems related to the construction and the structure of the Chebyshev recurrence are studied. Concurrently, the solutions of the Chebyshev recurrence lead to the notion of a formal Chebyshev series solution of a differential equation. A theorem describes the asymptotic behavior of coefficients of such a formal series which can be divergent. In some cases, the link between a divergent Chebyshev series and an actual fonction verifying the same differential equation can shown either by resummation methods or by a sequence of integrals in the complex plane.
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Contributor : Luc Rebillard <>
Submitted on : Thursday, February 24, 2005 - 11:51:29 AM
Last modification on : Friday, November 6, 2020 - 4:08:38 AM
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  • HAL Id : tel-00008571, version 1




Luc Rebillard. Etude théorique et algorithmique des séries de Chebyshev solutions d'équations différentielles holonomes. Mathématiques [math]. Institut National Polytechnique de Grenoble - INPG, 1998. Français. ⟨tel-00008571⟩



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