A recursive method for determining the one-dimensional submodules of Laurent-Ore modules

Abstract

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules. The hyperexponential solutions are computed recursively by solving a sequence of first-order ordinary matrix equations. As the recursion proceeds, the matrix equations will have constant coefficients with respect to the operators that have been considered.

Liouvillian solutions of linear difference–differential equations

“…(A 2 ) we can compute all hypergeometric solutions over C(x) of an ordinary difference equation σ(Y ) =ÂY where ∈ GL n (C(x)) ( [13,21,22,23,24,15]);…”

On Solutions of Linear Functional Systems and Factorization of Laurent–ore Modules

“…The first subproblem can be solved by a recursive method [15] for determining onedimensional submodules of a Laurent-Ore module. Applying the method to ∧ d M yields several finite subsets S 1 , .…”

“…Apply the algorithm in [15], we find that M has no one-dimensional submodules, so M is irreducible. Example 6.4 Let F, δ x , σ k be as given in Example 3.2 and L = F [∂ x , ∂ k , ∂ −1 k ] the Laurent-Ore algebra.…”

“…Clearly, form a basis of ∧ 2 M over F . By the algorithm in [15], every one-dimensional submodule of ∧ 2 M has a generator of the form w = (f 1 ,…”

“…, z nn ) τ . This system is clearly ∂-finite, so its rational solutions can be found by a specialized version of the method in [15]. A C-basis of all rational solutions of this system yields a C-basis {P 1 , .…”