Overview of the Ore_algebra Package - Maple Programming Help

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Overview of the Ore_algebra Package

 

Calling Sequence

Description

List of Ore_algebra Package Commands

Examples

Calling Sequence

Ore_algebra[command](arguments)

command(arguments)

Description

• 

The Ore_algebra package is a collection of commands for doing basic calculations in algebras of linear operators called Ore algebras.

• 

Examples of Ore algebras are algebras of differential operators, of shift or difference operators, and of q-dilation or q-difference operators, in each case with rational or polynomial coefficients.

• 

An Ore algebra is an algebra of noncommutative polynomials in the indeterminates x1,...,xp,D1,...,Dq ruled by the following commutation relations.

Dixj=σixjDi+δixj

  

Any other pair of indeterminates commute.  The sigma_is are algebra endomorphisms and the delta_is are module endomorphisms, with the additional property:

δipq=σipδiq+δipq

  

This has to be viewed as a skew variant of the Leibniz rule.

• 

Note that Ore algebras in the linear operators D1,...,Dp with polynomial coefficients in x1,...,xp reduce to Weyl algebras (differential case) when the sigma_i and the delta_i satisfy:

σixi=xi,``

δixi=1

  

(See Ore_algebra[Weyl_algebra].)

• 

Each command in the Ore_algebra package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

List of Ore_algebra Package Commands

• 

The following is a list of  available commands.

  

Building an algebra

diff_algebra

dual_algebra

poly_algebra

qshift_algebra

reverse_algebra

shift_algebra

skew_algebra

 

  

Calculations in an algebra

annihilators

dual_polynomial

rand_skew_poly

reverse_polynomial

skew_elim

skew_gcdex

skew_pdiv

skew_power

skew_prem

skew_product

 

 

  

Action on Maple objects

applyopr

 

 

 

  

Converters

Ore_to_DESol

Ore_to_diff

Ore_to_RESol

Ore_to_shift

  

To display the help page for a particular Ore_algebra command, see Getting Help with a Command in a Package.

• 

The skew_algebra, diff_algebra, shift_algebra, and poly_algebra commands declare new algebras to work with. They return a table needed by other Ore_algebra procedures. The skew_algebra command creates a general Ore algebra. The other commands are more specific and simplified alternatives to create algebras based on differential operators (see Weyl algebras), on shift or q-shift operators and commutative algebras of polynomials, respectively.

• 

The skew_product and skew_power commands implement the arithmetic of Ore algebras. Skew polynomials in an Ore algebra are represented by commutative polynomials of Maple.  The sum of skew polynomials is performed using the Maple `+` function. Their product, however, is performed using the skew_product command. Correspondingly, powers of skew polynomials are computed using the skew_power command.

• 

The rand_skew_poly command generates a random element of an Ore algebra.

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The applyopr command applies an operator of an Ore algebra to a function.

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The annihilators, skew_pdiv, skew_prem, skew_gcdex and skew_elim commands implement a skew Euclidean algorithm in Ore algebras and provide with related functionalities, such as computing remainders, gcds, (limited) elimination. The annihilators command makes it possible to compute a lcm of two skew polynomials. The skew_pdiv command computes pseudo-divisions in an Ore algebra, while skew_prem simply computes corresponding pseudo-remainders. The skew_gcdex command performs extended gcd computation in an Ore algebra. When possible, the skew_elim command eliminates an indeterminate between two skew polynomials.

Examples

withOre_algebra:

Compute the following sum of binomials:

k=0nbinomialn,k

k=0nbinomialn,k

(1)

Eliminate the index of summation k between operators that define binomialn,k.

Askew_algebrashift=Sn,n,δ=Dk,k,polynom=k:

elskew_elimn+1kSnn+1,k+1Dk+n2k+1,k,A:

This gives the following first-order recurrence satisfied by the sum.

cocollectprimpartsubsDk=0,el,Sn,Sn,factor

co:=Sn2

(2)

applyoprco,sn,A

2sn+sn+1

(3)

which is solved:

LREtools[hypergeomsols],sn,s0=1

2n

(4)

The sum of squares of binomials.

k=0nbinomialn,k2

k=0nbinomialn,k2

(5)

Eliminate the index of summation k between operators that define h.

elskew_elimn+1k2Snn+12,k+12Dkn+1n2k1,k,A:

This gives the following first-order recurrence satisfied by the sum.

cocollectprimpartsubsDk=0,el,Sn,Sn,factor

co:=n+2Sn2+4n6Sn

(6)

applyoprco,sn,A

n+2sn+2+4n6sn+1

(7)

which is solved:

LREtools[hypergeomsols],sn,s0=1,s1=2

4nΓn+12πΓn+1

(8)

See Also

Ore_algebra/Weyl_algebra

UsingPackages

with

 


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