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Calling Sequence
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MultiplyOperators(L1, L2, ...)
GCRD(L1, L2, ...)
LCLM(L1, L2, ...)
RightDivision(L1, L2)
RecurrenceToOperator(R, dvar)
OperatorToRecurrence(L, dvar)
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Parameters
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L1,L2, ...
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linear difference operators
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R
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recurrence
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dvar
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dependent variable
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Description
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The shift operator (often denoted as , , or ) acts on functions by adding +1 to the independent variable (often denoted as or ). If for example the shift operator is denoted with , and the independent variable by , then is the operator that sends an expression to .
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One can choose the name of the shift operator by assigning it to _Env_LRE_tau, and the name of the independent variable by assigning it to _Env_LRE_x. If these environment variables are assigned then they will be used to denote the shift operator and independent variable.
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An operator in [] can be written as = + ... + for rational functions in . If the dependent variable dvar is for example , then the equation is the recurrence relation + ... + = 0. So a difference operator represents a linear homogeneous recurrence relation. Converting representations can be done with the RecurrenceToOperator and OperatorToRecurrence commands.
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The product := corresponds to composition of linear difference operators. For example, if is the shift operator and is the independent variable, then will send any expression to , while the operator sends to . The product sends to = = while the product sends to = = . So acts the same as , which means that the operator equals the operator .
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If := and if is a solution of , in other words , then = = = . So right-factors of are important for solving because solutions of right-factors are also a solutions of (this is not true for left-factors, which is why GCLD/LCRM/LeftDivision are omitted here).
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The assignment := computes the Least Common Left Multiple of operators and , which means that and are right-factors of , and is minimal with this property. Then the solution space of is the sum of the solution spaces of and . The same functionality is provided by gfun[`rec+rec`]. Difference operators are also a special case of Ore operators.
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The assignment := computes the Greatest Common Right Divisor of and , which means that is a right-factor of both and , and is maximal with this property. Then the solution space of is the intersection of the solution spaces of and .
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One may specify more than two operators in MultiplyOperators, GCRD, or LCLM. For instance, := is the Least Common Left Multiple of , , , so solutions of are sums of solutions of , , and .
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The assignment := right-divides by . This means that where the order of is less than that of . will be if and only if is a right-factor of .
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Examples
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>
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>
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>
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>
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R is not zero so L3 is not a right-factor of L
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| (9) |
>
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>
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| (12) |
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Compatibility
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The LREtools[MultiplyOperators], LREtools[GCRD], LREtools[LCLM], LREtools[RightDivision], LREtools[RecurrenceToOperator] and LREtools[OperatorToRecurrence] commands were introduced in Maple 2021.
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