Apparent singularities of linear difference equations with polynomial coefficients

Abstract

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

Liouvillian solutions of irreducible linear difference equations

“…Note: this definition of apparent singularity is related, but not quite equivalent, to the definition in [1].…”

“…(1) The set of all such linear difference operators is denoted by C(x) [τ ]. A solution of L is a function u which satisfies equation (1). In this paper we aim to find Liouvillian solutions for linear difference equations.…”

Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients

“…In the scalar case, several desingularization algorithms exist for differential, difference (e.g., [2]), and more generally, Ore operators (see, e.g. [11,10] and references therein).…”

Analytic Solutions of Linear Difference Equations, Formal Series, and Bottom Summation

“…where R ∈ C[k, E k ] exists, equality (2) gives an opportunity to use the discrete Newton-Leibniz formula…”

“…This type of summation we call bottom summation. Some important auxillary statements (Section 2) on sequences of power series are based on the idea of the ε-deformation of a difference operator which was first used by M. van Hoeij in [7]; later this idea was used in [4] and in [2] as well.…”