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Ore_algebra

  

shift_algebra

  

create an algebra of linear difference operators

  

qshift_algebra

  

create an algebra of linear q-difference operators

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

shift_algebra(l_1, ..., l_n)

qshift_algebra(lq_1, ..., lq_n)

Parameters

l_i

-

list Si,ni or list comm,ai

lq_i

-

list Si,ni,qi or list comm,ai

S_i

-

indeterminates (shift and q-shift operator names)

n_i

-

indeterminates (variable names)

a_i

-

indeterminates (parameter names)

Description

• 

The shift_algebra(l_1, ..., l_n) and qshift_algebra(lq_1, ..., lq_n) functions each declare an Ore algebra and return a table that is used by other functions of the Ore_algebra package.

• 

A difference algebra is an algebra of noncommutative polynomials in the indeterminates n1,...,np,S1,...,Sp ruled by the following commutation relations:

Sini=ni+1Si

  

for i=1,...,p.  Any other pair of indeterminates commute.

• 

A q-difference algebra is an algebra of noncommutative polynomials in the indeterminates q,n1,...,np,S1,...,Sp ruled by the following commutation relations:

Sini=qni+1Si

  

for i=1,...,p.  q is a constant and any other pair of indeterminates commute.

  

Note: Difference and q-difference algebras are special cases of Ore algebras. For more information, see Ore_algebra.

• 

The name n_i can be unassigned.

• 

The name S_i can be unassigned.  It is used to denote the difference or q-difference indeterminate S_i associated to the base indeterminate n_i, that is, the operator of shift or q-shift with respect to n_i.

• 

When the list l_i is of the form Si,ni (difference case) or Si,ni,qi (q-difference case), the names n_i and S_i can be unassigned.  Both indeterminates commute with any other indeterminate of the algebra.

• 

When the list l_i is of the form comm,ai, the name a_i can be unassigned.  It denotes a parameter that commutes with any other indeterminate of the algebra.

• 

Though difference and q-difference algebras are noncommutative algebras, their elements are represented with the standard commutative Maple product.  Every Ore_algebra function dealing with elements of a difference of q-difference algebra uses its normal form where all S_i appear on the right of the corresponding n_i.  A monomial naSnb can therefore be printed either naSnb or naSnb.

• 

The sum in difference or q-difference algebras is performed by simply using the Maple `+`, while the product is performed by the Ore_algebra function skew_product (see examples below).

• 

It is also possible to declare a difference or a q-difference algebra by using Ore_algebra[skew_algebra].  Moreover, the algebras declared by Ore_algebra[shift_algebra] and Ore_algebra[qshift_algebra] are difference and q-difference algebras based on shift and q-shift operators S_i, but it is also possible to declare algebras based on finite difference and q-difference operators Di=Si+1 (see Ore_algebra[skew_algebra], predefined types delta and qdelta).

• 

Options are available to control the ground ring of the algebra and the action of the operators on Maple objects.  See Ore_algebra[declaration_options].

• 

These function are part of the Ore_algebra package, and so can be used in the form shift_algebra(..) and qshift_algebra(..) only after performing the command with(Ore_algebra) or with(Ore_algebra,<function>).  The functions can always be accessed in the long form Ore_algebra[shift_algebra](..) and Ore_algebra[qshift_algebra](..).

Examples

withOre_algebra&colon;

Difference algebras:

Ashift_algebraSn&comma;n&comma;Sm&comma;m

A:=Ore_algebra

(1)

skew_productSn&comma;n&comma;A&comma;skew_productSm&comma;m&comma;A

n&plus;1Sn&comma;m&plus;1Sm

(2)

skew_productSnSm&comma;nm&comma;A

mn&plus;m&plus;n&plus;1SnSm

(3)

skew_productSn&comma;n10&comma;A

n10&plus;10n9&plus;45n8&plus;120n7&plus;210n6&plus;252n5&plus;210n4&plus;120n3&plus;45n2&plus;10n&plus;1Sn

(4)

Both following calls are equivalent.  The first syntax is more convenient to input numerous commutative parameters.

skew_algebracomm&equals;a&comma;b&comma;c&comma;d&comma;e&comma;f&comma;g&comma;h&comma;shift&equals;Sn&comma;n

Ore_algebra

(5)

shift_algebracomm&comma;a&comma;comm&comma;b&comma;comm&comma;c&comma;comm&comma;d&comma;comm&comma;e&comma;comm&comma;f&comma;comm&comma;g&comma;comm&comma;h&comma;Sn&comma;n

Ore_algebra

(6)

evalb&equals;

true

(7)

Both following algebras are different points of view for the same algebra of operators

shift_algebraSn&comma;n

Ore_algebra

(8)

(or equivalently skew_algebra(shift=[Sn, n]);).

skew_algebra&delta;&equals;Dn&comma;n

Ore_algebra

(9)

q-difference algebras:

Aqshift_algebraSn&comma;qn&comma;Sm&comma;qm

A:=Ore_algebra

(10)

skew_productSn&comma;qn&comma;A&comma;skew_productSm&comma;qm&comma;A

qqnSn&comma;qqmSm

(11)

skew_productSnSm&comma;qnqm&comma;A

q2qnqmSnSm

(12)

skew_productSn&comma;qn10&comma;A

q10qn10Sn

(13)

There can also be distinct qs.

Aqshift_algebraSn&comma;qn&comma;Sm&comma;pm

A:=Ore_algebra

(14)

skew_productSn&comma;qn&comma;A&comma;skew_productSm&comma;pm&comma;A

qqnSn&comma;ppmSm

(15)

skew_productSnSm&comma;qnpm&comma;A

qqnppmSnSm

(16)

skew_productSn&comma;qn10&comma;A

q10qn10Sn

(17)

See Also

Ore_algebra

Ore_algebra/skew_product

 


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