Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

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• 作者：S.A. Abramov^a ;   ; ¹ ;   ; ^{sabramov@ccas.ru} ;   ; ^{sergeyabramov@mail.ru} ; M.A. Barkatou^b ;   ; ^{moulay.barkatou@unilim.fr} ; M. van Hoeij^c ;   ; ^{hoeij@math.fsu.edu} ; M. Petkovš ; ek^d ;   ; ^{Marko.Petkovsek@fmf.uni-lj.si}
• 关键词：Entire solutions of linear difference equations ; Subanalytic solutions of linear recurrence equations ; Hypergeometric sequences
• 刊名：Journal of Symbolic Computation
• 出版年：2011
• 出版时间：November 2011
• 年：2011
• 卷：46
• 期：11
• 页码：1205-1228
• 全文大小：357 K

文摘

We consider linear difference equations with polynomial coefficients over and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the -linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to of entire solutions and that the dimension of this space is equal to the order of the original equation.

We also consider -dimensional () hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most , and that this dimension may be equal to for some systems (although the dimension of the space of all sequential solutions is always positive).

Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.