Overview - Maple Help

Overview of the Ore_algebra Package

 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 ${x}_{1},...,{x}_{p},{\mathrm{D}}_{1},...,{\mathrm{D}}_{q}$ ruled by the following commutation relations.

${\mathrm{D}}_{i}{x}_{j}={\mathrm{\sigma }}_{i}\left({x}_{j}\right){\mathrm{D}}_{i}+{\mathrm{\delta }}_{i}\left({x}_{j}\right)$

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

${\mathrm{\delta }}_{i}\left(pq\right)={\mathrm{\sigma }}_{i}\left(p\right){\mathrm{\delta }}_{i}\left(q\right)+{\mathrm{\delta }}_{i}\left(p\right)q$

 This has to be viewed as a skew variant of the Leibniz rule.
 • Note that Ore algebras in the linear operators ${\mathrm{D}}_{1},...,{\mathrm{D}}_{p}$ with polynomial coefficients in ${x}_{1},...,{x}_{p}$ reduce to Weyl algebras (differential case) when the sigma_i and the delta_i satisfy:

${\mathrm{\sigma }}_{i}\left({x}_{i}\right)={x}_{i},$

${\mathrm{\delta }}_{i}\left({x}_{i}\right)=1$

 • 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

 Calculations in an algebra

 Action on Maple objects

 Converters

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

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$

Compute the following sum of binomials:

 > $\mathrm{Sum}\left(\mathrm{binomial}\left(n,k\right),k=0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right)$ (1)

Eliminate the index of summation k between operators that define $\left(\genfrac{}{}{0}{}{n}{k}\right)$.

 > $A≔\mathrm{skew_algebra}\left(\mathrm{shift}=\left[\mathrm{Sn},n\right],\mathrm{\delta }=\left[\mathrm{Dk},k\right],\mathrm{polynom}=k\right):$
 > $\mathrm{el}≔\mathrm{skew_elim}\left(\left(n+1-k\right)\mathrm{Sn}-\left(n+1\right),\left(k+1\right)\mathrm{Dk}+\left(n-2k+1\right),k,A\right):$

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

 > $\mathrm{co}≔\mathrm{collect}\left(\mathrm{primpart}\left(\mathrm{subs}\left(\mathrm{Dk}=0,\mathrm{el}\right),\mathrm{Sn}\right),\mathrm{Sn},\mathrm{factor}\right)$
 ${\mathrm{co}}{≔}{\mathrm{Sn}}{-}{2}$ (2)
 > $\mathrm{applyopr}\left(\mathrm{co},s\left(n\right),A\right)$
 ${-}{2}{}{s}{}\left({n}\right){+}{s}{}\left({n}{+}{1}\right)$ (3)

which is solved:

 > $\mathrm{LREtools}\left[\mathrm{hypergeomsols}\right]\left(,s\left(n\right),\left\{s\left(0\right)=1\right\}\right)$
 ${{2}}^{{n}}$ (4)

The sum of squares of binomials.

 > $\mathrm{Sum}\left({\mathrm{binomial}\left(n,k\right)}^{2},k=0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right)}^{{2}}$ (5)

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

 > $\mathrm{el}≔\mathrm{skew_elim}\left({\left(n+1-k\right)}^{2}\mathrm{Sn}-{\left(n+1\right)}^{2},{\left(k+1\right)}^{2}\mathrm{Dk}-\left(n+1\right)\left(n-2k-1\right),k,A\right):$

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

 > $\mathrm{co}≔\mathrm{collect}\left(\mathrm{primpart}\left(\mathrm{subs}\left(\mathrm{Dk}=0,\mathrm{el}\right),\mathrm{Sn}\right),\mathrm{Sn},\mathrm{factor}\right)$
 ${\mathrm{co}}{≔}\left({n}{+}{2}\right){}{{\mathrm{Sn}}}^{{2}}{+}\left({-}{4}{}{n}{-}{6}\right){}{\mathrm{Sn}}$ (6)
 > $\mathrm{applyopr}\left(\mathrm{co},s\left(n\right),A\right)$
 $\left({n}{+}{2}\right){}{s}{}\left({n}{+}{2}\right){+}\left({-}{4}{}{n}{-}{6}\right){}{s}{}\left({n}{+}{1}\right)$ (7)

which is solved:

 > $\mathrm{LREtools}\left[\mathrm{hypergeomsols}\right]\left(,s\left(n\right),\left\{s\left(0\right)=1,s\left(1\right)=2\right\}\right)$
 $\frac{{{4}}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{1}}{{2}}\right)}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({n}{+}{1}\right)}$ (8)