convert a linear ordinary differential equation to an OrePoly structure
convert a linear recurrence equation to an OrePoly structure
convert an OrePoly structure to a linear ordinary differential equation
convert an OrePoly structure to a linear recurrence equation
convert a FactoredOrePoly structure to a linear ordinary differential equation
convert a FactoredOrePoly structure to a linear recurrence equation
convert a FactoredOrePoly structure to a OrePoly structure
left hand side of a linear equation (either differential or recurrence)
function from eq, for example, f(x)
factored Ore operator
name of the independent variable
parameter indicating the case of the equation ('differential' or 'shift')
The LinearOperators[DEToOrePoly] and LinearOperators[REToOrePoly] functions return an Ore operator K such that eq = K(f). The LinearOperators[OrePolyToDE], LinearOperators[OrePolyToRE], LinearOperators[FactoredOrePolyToDE], and LinearOperators[FactoredOrePolyToRE] functions apply the operator (L or M) to the function f. The LinearOperators[FactoredOrePolyToOrePoly] function converts an Ore polynomial in factored form, that is, a FactoredOrePoly structure, to an Ore polynomial in expanded form, that is, an OrePoly structure.
A completely factored Ore operator is represented by a structure that consists of the keyword FactoredOrePoly and a sequence of lists. Each list consists of two elements and describes a first degree factor. The first element provides the zero degree coefficient and the second element provides the first degree coefficient. For example, in the differential case with a differential operator D, FactoredOrePoly([-1, x], [x, 0], [4, x^2], [0, 1]) describes the operator −1+xD⁡x⁡x2⁢D+4⁡D.
An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+x⁢D+x+1⁢D2+D3.
poly ≔ FactoredOrePoly⁡x5,1+x,x,−1
ode ≔ LinearOperatorsFactoredOrePolyToDE⁡poly,y⁡x
lre ≔ LinearOperatorsFactoredOrePolyToRE⁡poly,y⁡x
lre ≔ LinearOperatorsOrePolyToRE⁡OrePoly⁡x,x3+x−2,x5,y⁡x
L ≔ FactoredOrePoly⁡32⁢x,1,12⁢x,1,−12⁢x,1
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