Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients

Abstract

Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a non-zero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic extension of a transcendental liouvillian extension of the field of rational functions. We show how one can use this result to give a procedure to find a basis for the space of solutions, liouvillian over K, of L(y)=0 where K is such a field and L(y) has coefficients in K.

Rational Solutions of Riccati-like Partial Differential Equations

“…We denote by T andT the solutions of (4.8) in F(t) andF(t), respectively. Algorithms by Singer (1991) and Bronstein (1992a) are able to computeT. We present a theorem to describe the structure of T and modify Bronstein's algorithm solve riccati to compute T.…”

“…For example, solutions h with the property that the quotient p = h /h ∈ C(x) may be represented as h = exp p dx if p satisfies the first-order Riccati equation p + p 2 + ap + b = 0. Equivalently, this linear ODE allows the first-order right factor y − qy over C(x) if q obeys the same equation as p. In general, finding the first-order right rational factors of a linear homogeneous ODE is equivalent to finding the rational solutions of its associated Riccati equation (see, for example, Singer, 1991).…”

Solving parameterized linear difference equations in terms of indefinite nested sums and products

“…Another approach is [23] where one can try to solve problem PLDE for a subclass of monomial extensions that covers besides indefinite nested products (Π-extensions) also differential fields; see also [24]. The only restriction is that one cannot consider indefinite nested sums and products (ΠΣ-extensions) that arise frequently in symbolic summation.…”

On Solutions of Linear Ordinary Difference Equations in their Coefficient Field

“…This is the problem that we address in this paper, namely, given a difference field k with its automorphism σ, g ∈ k and a linear ordinary difference operator L with coefficients in k, to compute all the solutions in k of the equation Ly = g. There are known solutions to this problem when k = C(x) and σ is the automorphism over C given by σx = x + 1 (Abramov, 1989) or σx = qx (Abramov, 1995), but no generalization to other automorphisms or more general coefficient fields. In the theory of linear ordinary differential equations, the concepts of Liouvillian (Singer, 1991) and monomial (Bronstein, 1990(Bronstein, , 1997 extensions of differential fields have led to extensions of rational techniques that solve a similar problem with more general functions allowed in the coefficients (Singer, 1991;Bronstein, 1992). In the case of difference fields, Karr introduced ΠΣ-fields and used them to develop summation algorithms that allow more general summands (Karr, 1981(Karr, , 1985.…”