Hypergeometric solutions of linear difference systems

Abstract

We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $\tau \left(Y\right)=MY$, with $M\in {\mathrm{GL}}_n\left(C\right(x\left)\right)$, where τ is the shift operator. Hypergeometric solutions are solutions of the form γP where $P\in C{\left(x\right)}^n$ and γ is a hypergeometric term over $C\left(x\right)$, i.e. $\tau \left(\gamma \right)/\gamma \in C\left(x\right)$. Our contributions concern efficient computation of a set of candidates for $\tau \left(\gamma \right)/\gamma$ which we write as $\lambda =c\frac{A}{B}$ with monic $A,B\in C\left[x\right]$, $c\in C^{⁎}$. Factors of the denominators of $M^{-1}$ and M give candidates for A and B, while another algorithm is needed for c. We use 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 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.