Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences
Résumé
We consider lineardifferenceequations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalyticsolutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation. We also consider d-dimensional (d≥1) hypergeometricsequences, i.e., sequential and subanalyticsolutions of consistent systems of first-order differenceequations for a single unknown function. We show that the dimension of the space of subanalyticsolutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive). Subanalyticsolutions 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 subanalyticsolution of an equation or a system, but can give incorrect results for some sequential solutions.