Hypergeometric solutions of linear difference systems
M Barkatou, M van Hoeij, J Middeke, Y Zhou - arXiv preprint arXiv …, 2024 - arxiv.org
M Barkatou, M van Hoeij, J Middeke, Y Zhou
arXiv preprint arXiv:2401.08470, 2024•arxiv.orgWe extend Petkov\v {s} ek's algorithm for computing hypergeometric solutions of scalar
difference equations to the case of difference systems $\tau (Y)= MY $, with $ M\in {\rm GL}
_n (C (x)) $, where $\tau $ is the shift operator. Hypergeometric solutions are solutions of the
form $\gamma P $ where $ P\in C (x)^ n $ and $\gamma $ is a hypergeometric term over $ C
(x) $, ie ${\tau (\gamma)}/{\gamma}\in C (x) $. Our contributions concern efficient
computation of a set of candidates for ${\tau (\gamma)}/{\gamma} $ which we write as …
difference equations to the case of difference systems $\tau (Y)= MY $, with $ M\in {\rm GL}
_n (C (x)) $, where $\tau $ is the shift operator. Hypergeometric solutions are solutions of the
form $\gamma P $ where $ P\in C (x)^ n $ and $\gamma $ is a hypergeometric term over $ C
(x) $, ie ${\tau (\gamma)}/{\gamma}\in C (x) $. Our contributions concern efficient
computation of a set of candidates for ${\tau (\gamma)}/{\gamma} $ which we write as …
We extend Petkov\v{s}ek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems , with , where is the shift operator. Hypergeometric solutions are solutions of the form where and is a hypergeometric term over , i.e. . Our contributions concern efficient computation of a set of candidates for which we write as with monic , . Factors of the denominators of and give candidates for and , while another algorithm is needed for . We use the super-reduction algorithm to compute candidates for , as well as other ingredients to reduce the list of candidates for . To further reduce the number of candidates , we bound the so-called type of by bounding local types. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.
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