On m-Interlacing Solutions of Linear Difference Equations

Authors:
S. A. Abramov;M. A. Barkatou;D. E. Khmelnov
Affiliations:
Computing Centre of the Russian Academy of Sciences, Moscow, Russia 119991,GSP-1;Institute XLIM, Université de Limoges, CNRS, Limoges cedex, France 87060;Computing Centre of the Russian Academy of Sciences, Moscow, Russia 119991,GSP-1
Venue:
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Year:
2009

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Abstract

We consider linear homogeneous difference equations with rational-function coefficients. The search for solutions in the form of the m -interlacing ($1\leq m\leq {\mathop{\rm ord}} L$, where L is a given operator) of finite sums of hypergeometric sequences, plays an important role in the Hendriks---Singer algorithm for constructing all Liouvillian solutions of L (y ) = 0. We show that Hendriks---Singer's procedure for finding solutions in the form of such m -interlacing can be simplified. We also show that the space of solutions of L (y ) = 0 spanned by the solutions of the form of the m -interlacing of hypergeometric sequences possesses a cyclic permutation property. In addition, we describe adjustments of our implementation of the Hendriks---Singer algorithm to utilize the presented results.