Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over CC and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the CC-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to ZZ of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to ZZ of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider dd-dimensional (d≥1d≥1) 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 11, and that this dimension may be equal to 00 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.

► Linear difference equations with polynomial coefficients over CC are considered.
► A subanalytic solution is the restriction of an analytic solution to ZZ.
► The dimension of the space of subanalytic solutions equals the order of the equation.
► The dimension of subanalytic solutions of a hypergeometric system is 1 or 0.
► Certain summation algorithms are guaranteed to work correctly on subanalytic inputs.