The paper investigates the problem of converting holonomic sequences, which are solutions to linear difference equations, into simple rational recursive sequences, which are solutions to rational recurrence relations.
The key insights are:
Proposition 1 shows that every holonomic sequence of degree 1 can be converted to a simple rational recursive sequence of order at most one higher than the original holonomic sequence.
Proposition 2 extends this result to holonomic sequences of degree up to 3, showing they can be converted to simple rational recursive sequences of order at most two higher than the original.
Theorem 2 provides a general result, proving that every holonomic sequence of order l and degree d can be converted to a simple rational recursive sequence of order at most l+d.
The paper presents two algorithms for this conversion. The Gröbner bases method iteratively computes elimination ideals to find a simple rational recursive generator of minimal order. The linear algebra method directly constructs the simple rational recursive equation based on the proof of Theorem 2, sacrificing minimality of order for better efficiency.
The authors demonstrate the application of these algorithms on several examples, including converting polynomial and C-finite sequences to simple rational recursive form. The results show that holonomic sequences can be efficiently converted to a more restricted class of sequences with a rational recursive structure.
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by Bertrand Teg... at arxiv.org 05-01-2024
https://arxiv.org/pdf/2404.19136.pdfDeeper Inquiries