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LinearOperators

  

Apply

  

apply an Ore polynomial to a function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Apply(L,expr, x, case)

Parameters

L

-

Ore polynomial

expr

-

algebraic expression

x

-

name of the independent variable

case

-

parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

The LinearOperators[Apply] function applies the polynomial L to the expression expr. The polynomial can be an OrePoly or a FactoredOrePoly 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+xDxDx2+4D.

• 

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+xD+x+1D2+D3.

• 

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

• 

This command is part of the LinearOperators package, so it can be used in the form Apply(..) only after executing the command with(LinearOperators).  However, it can always be used in the form LinearOperators[Apply](..).

Examples

exprⅇxx1+x2+1

expr:=ⅇxx1+x2+1

(1)

LLinearOperators[FactoredAnnihilator]expr,x,'differential'

L:=FactoredOrePolyx4+6x22x+3x3x2+x+1x2+1x1,1,1x12,1

(2)

LinearOperators[Apply]L,expr,x,'differential'

0

(3)

References

  

Abramov, S. A., and Zima, E. V. "Minimal Completely Factorable Annihilators." In Proceedings of ISSAC '97, pp. 290-297. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.

See Also

LinearOperators

LinearOperators[converters]

LinearOperators[FactoredAnnihilator]

 


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