Let K be a field of characteristic zero and M(Y ) = N a system of linear differential equations with coefficients in K(x). We propose a new algorithm to compute the set of rational solutions of such a system. This algorithm does not require the use of cyclic vectors. It has been implemented in Maple V and it turns out to be faster than cyclic vector computations. We show how one can use this algorithm to give a method to find the set of solutions with entries in K(x)[log x] of M(Y ) = N .
We propose a method for computing the regular singular formal solutions of a linear differential system in the neighbourhood of a singular point. This algorithm avoids the use of cyclic vectors and has been implemented † in the computer algebra system Maple.
Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L. We prove that one can effectively construct a left multiple with polynomial coefficientsL of L such that every singularity ofL is a singularity of L that is not apparent. As a consequence, if all singularities of L are apparent, then L has a left multiple whose trailing and leading coefficients equal 1.In the differential case, takewhere d is the order of L (so a d (z) = 0) and ∂ = d/dz. Let S(L) ⊂ C be the set of roots of a d (z). The finite singularities of L are the elements of S(L). The singularity at ∞ will not be considered in this paper. If p ∈ S(L) and if there exist d linearly independent analytic solutions at z = p then p is called an apparent singularity. Suppose L has apparent singularities. The question is if it is possible to construct another operatorL ∈ C[z, ∂] of higher order such that any solution of L(y) = 0 is a solution ofL(y) = 0, and S(L) = {p ∈ S(L) | p not apparent}. In the differential case the answer is affirmative, see [10]. In this paper we give the affirmative answer to the corresponding question for the difference case.In the remainder of this paper (except the appendix) only the difference case will be considered. The shift operator E acts on functions of the complex variable z as Ey(z) = y(z + 1). We consider non-commutative operator rings C[z, E] and C(z)[E] (the rings of linear difference operators with polynomial and, resp., rational function coefficients over C). Let(1) Assume that the leading coefficient a d (z) and the trailing coefficient a 0 (z) are both non-zero, and that a 0 (z), . . . , a d (z) do not have a non-constant common factor. Set ord L = d.Definition 1 A root p of a 0 (z) is called a t-singularity (a trailing singularity). A root p of a d (z − d) is called an l-singularity (a leading singularity).Definition 2 A right-holomorphic (resp. left-holomorphic) function is a meromorphic function on C that is holomorphic on some right (resp. left) half plane. In other words, holomorphic when Re z (resp. − Re z) is sufficiently large. A half-holomorphic function is a function that is right-or left-holomorphic.Definition 3 A root p of a 0 (z) (resp. of a d (z − d)) is called an apparent t-(resp. l)-singularity if no right-(resp. left)-holomorphic solution has a pole at p. An operatorL is a t-(resp. l)-desingularization of L if every meromorphic solution of L is a solution ofL, and every t-(resp. l)-singularity of L is a t-(resp. l)-singularity of L that is not apparent.We show that both t-and l-desingularizations exist. We give algorithms t-desing and l-desing for constructing a t-(resp. l)-desingularization and
We propose an algorithm to compute rational function solutions for a rst order system of linear di erence equations with rational coe cients. This algorithm does not require preliminary uncoupling of the given system.
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