Hypergeometric Solutions of Linear Difference Systems
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908281781354496 |
|---|---|
| author | Barkatou, Moulay van Hoeij, Mark Middeke, Johannes Zhou, Yi |
| author_facet | Barkatou, Moulay van Hoeij, Mark Middeke, Johannes Zhou, Yi |
| contents | We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $τ(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $τ$ is the shift operator. Hypergeometric solutions are solutions of the form $γP$ where $P \in C(x)^n$ and $γ$ is a hypergeometric term over $C(x)$, i.e. ${τ(γ)}/γ \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${τ(γ)}/γ$ which we write as $λ= c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypergeometric Solutions of Linear Difference Systems Barkatou, Moulay van Hoeij, Mark Middeke, Johannes Zhou, Yi Symbolic Computation 39A04 G.2.1 We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $τ(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $τ$ is the shift operator. Hypergeometric solutions are solutions of the form $γP$ where $P \in C(x)^n$ and $γ$ is a hypergeometric term over $C(x)$, i.e. ${τ(γ)}/γ \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${τ(γ)}/γ$ which we write as $λ= c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ 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. |
| title | Hypergeometric Solutions of Linear Difference Systems |
| topic | Symbolic Computation 39A04 G.2.1 |
| url | https://arxiv.org/abs/2401.08470 |