Computing Hypergeometric Solutions of Linear Recurrence Equations

Liouvillian solutions of irreducible linear difference equations

“…Apart from this proof and definition 6, everything else in this section and its subsections comes from [9], [7], [10] and [13]. The main new results in this paper are Theorems 2 and 3 in Section 3 because those are the results that make our algorithm efficient.…”

“…The statement in Theorem 1 in subsection 2.2 below was already mentioned in [7,Section 4] but a proof was not given there so we will give a proof in this paper. Apart from this proof and definition 6, everything else in this section and its subsections comes from [9], [7], [10] and [13].…”

“…55]. Computing Liouvillian solutions is already solved in [9,11,2,8] where this problem is reduced to computing hypergeometric solutions of some other operatorL (or system, in [8]) which has n times more finite singularities (defined in [10]) than the original operator L. The problem is that computing a hypergeometric solution is done with a combinatorial algorithm [12,7] where the number of combinations depends exponentially on the number of singularities. We give a more direct approach, based on Theorems 2 and 3, that avoids introducingL and the corresponding increase in the number of singularities.…”

“…In essence, the efficiency problem in the existing algorithm for Liouvillian solutions is this: If one computes hypergeometric solutions ofL (denoted Pi in [9,Lemma 5.3]), and if one does not exploit the fact that the singularities ofL come from singularities of L, then the algorithms for computing hypergeometric solutions [7,12] will try many more combinations than necessary. The contribution of this paper is to solve this problem with Theorems 2 and 3.…”

“…In Section 2, we review definitions and some properties of gauge transformations and valuation growths from [9], [7] and [10]. In Section 3, we define n-equivalence and show how it is related to gauge equivalence.…”