multiply linear difference operators
Greatest Common Right Divisor of linear difference operators
Least Common Left Multiple of linear difference operators
divide two linear difference operators
convert a recurrence relation to a difference operator
convert a difference operator to a recurrence relation
MultiplyOperators(L1, L2, ...)
GCRD(L1, L2, ...)
LCLM(L1, L2, ...)
linear difference operators
The shift operator (often denoted as E, S, or τ) acts on functions by adding +1 to the independent variable (often denoted as n or x). If for example the shift operator is denoted with E, and the independent variable by x, then E is the operator that sends an expression u⁡x to u⁡x+1.
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.
An operator L in C⁡x[E] can be written as L = an⁡x⁢En + ... + a0⁡x for rational functions ai⁡x in C⁡x. If the dependent variable dvar is for example u⁡x, then the equation L⁡u⁡x=0 is the recurrence relation an⁡x⁢u⁡x+n + ... + a0⁡x⁢u⁡x = 0. So a difference operator represents a linear homogeneous recurrence relation. Converting representations can be done with the RecurrenceToOperator and OperatorToRecurrence commands.
The product L := MultiplyOperators⁡L1,L2 corresponds to composition of linear difference operators. For example, if E is the shift operator and x is the independent variable, then E will send any expression u⁡x to u⁡x+1, while the operator x sends u⁡x to x⁢u⁡x. The product x⁢E sends u⁡x to x⁡E⁡u⁡x = x⁡u⁡x+1 = x⁢u⁡x+1 while the product E x sends u⁡x to E⁡x⁡u⁡x = E⁡x⁢u⁡x = x+1⁢u⁡x+1. So E x acts the same as x+1⁢E, which means that the operator E x equals the operator x+1⁢E.
If L := MultiplyOperators⁡L1,L2 and if u⁡x is a solution of L2, in other words L2⁡u⁡x=0, then L⁡u⁡x = L1⁡L2⁡u⁡x = L1⁡0 = 0. So right-factors of L are important for solving L because solutions of right-factors are also a solutions of L (this is not true for left-factors, which is why GCLD/LCRM/LeftDivision are omitted here).
The assignment L := LCLM⁡L1,L2 computes the Least Common Left Multiple of operators L1 and L2, which means that L1 and L2 are right-factors of L, and L is minimal with this property. Then the solution space of L is the sum of the solution spaces of L1 and L2. The same functionality is provided by gfun[`rec+rec`]. Difference operators are also a special case of Ore operators.
The assignment L := GCRD⁡L1,L2 computes the Greatest Common Right Divisor of L1 and L2, which means that L is a right-factor of both L1 and L2, and is maximal with this property. Then the solution space of L is the intersection of the solution spaces of L1 and L2.
One may specify more than two operators in MultiplyOperators, GCRD, or LCLM. For instance, L := LCLM⁡L1,L2,L3 is the Least Common Left Multiple of L1, L2, L3, so solutions of L are sums of solutions of L1, L2, and L3.
The assignment Q,R := RightDivision⁡L1,L2 right-divides L1 by L2. This means that L1=Q⁢L2+R where the order of R is less than that of L2. R will be 0 if and only if L2 is a right-factor of L1.
_Env_LRE_tau ≔ E;_Env_LRE_x ≔ x
L1 ≔ E2−x
L2 ≔ E2−E−x
L ≔ LCLM⁡L1,L2
L3 ≔ MultiplyOperators⁡E−x,L1
Q,R ≔ RightDivision⁡L,L3
R is not zero so L3 is not a right-factor of L
Lnew ≔ MultiplyOperators⁡Q,L3+R
Op ≔ RecurrenceToOperator⁡u⁡n+3+2⁢u⁡n+1+5⁢n⁢u⁡n=0,u⁡n
rec ≔ OperatorToRecurrence⁡Op,u⁡n
The LREtools[MultiplyOperators], LREtools[GCRD], LREtools[LCLM], LREtools[RightDivision], LREtools[RecurrenceToOperator] and LREtools[OperatorToRecurrence] commands were introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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