D’Alembertian series solutions at ordinary points of LODE with polynomial coefficients

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• 作者：S.A. Abramov ; M.A. Barkatou
• 关键词：Linear differential equations ; Linear recurrence equations ; Closed form solutions ; D’ ; Alembertian series ; Hypergeometric series
• 刊名：Journal of Symbolic Computation
• 出版年：2009
• 出版时间：January 2009
• 年：2009
• 卷：44
• 期：1
• 页码：48-59
• 全文大小：636 K

文摘

By definition, the coefficient sequence of a d’Alembertian series — Taylor’s or Laurent’s — satisfies a linear recurrence equation with coefficients in and the corresponding recurrence operator can be factored into first-order factors over (if this operator is of order 1, then the series is hypergeometric). Let L be a linear differential operator with polynomial coefficients. We prove that if the expansion of an analytic solution u(z) of the equation L(y)=0 at an ordinary (i.e., non-singular) point of L is a d’Alembertian series, then the expansion of u(z) is of the same type at any ordinary point. All such solutions are of a simple form. However the situation can be different at singular points.