skew_power - Maple Help

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Ore_algebra

 skew_product
 inner product of an Ore algebra
 skew_power
 power of an Ore algebra

 Calling Sequence skew_product(w1, w2, A) skew_power(w, n, A)

Parameters

 w1, w2, w - skew polynomials of the Ore algebra A A - Ore algebra table n - non-negative integer

Description

 • The skew_product(w1, w2, A) function computes the noncommutative product $\mathrm{w1}\mathrm{w2}$ (in this order) in the Ore algebra A.
 • The skew_power(w, n, A) function computes the nth power of w in the Ore algebra A.
 • To declare an Ore algebra, use Ore_algebra[skew_algebra], Ore_algebra[shift_algebra], or Ore_algebra[diff_algebra].
 • These functions are part of the Ore_algebra package, and so can be used in the form skew_product(..) only after performing the command with(Ore_algebra) or with(Ore_algebra,skew_product).  The functions can always be accessed in the long form Ore_algebra[skew_product](..).

Examples

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

Algebras of difference operators:

 > $A≔\mathrm{shift_algebra}\left(\left[\mathrm{Sn},n\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Sn},n,A\right)$
 $\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (1)
 > $\mathrm{skew_product}\left({\mathrm{Sn}}^{5},{n}^{3},A\right)$
 $\left({{n}}^{{3}}{+}{15}{}{{n}}^{{2}}{+}{75}{}{n}{+}{125}\right){}{{\mathrm{Sn}}}^{{5}}$ (2)

Algebras of differential operators:

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right),\mathrm{skew_product}\left(\mathrm{Dy},y,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}{,}{\mathrm{Dy}}{}{y}{+}{1}$ (3)
 > $\mathrm{skew_product}\left(\mathrm{Dx}\mathrm{Dy},xy,A\right)$
 ${\mathrm{Dx}}{}{\mathrm{Dy}}{}{x}{}{y}{+}{\mathrm{Dx}}{}{x}{+}{\mathrm{Dy}}{}{y}{+}{1}$ (4)
 > $\mathrm{skew_product}\left(\mathrm{Dx},{x}^{10},A\right)$
 ${\mathrm{Dx}}{}{{x}}^{{10}}{+}{10}{}{{x}}^{{9}}$ (5)
 > $\mathrm{skew_power}\left(\mathrm{Dx}+x,4,A\right)$
 ${{\mathrm{Dx}}}^{{4}}{+}{4}{}{x}{}{{\mathrm{Dx}}}^{{3}}{+}\left({6}{}{{x}}^{{2}}{+}{6}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({4}{}{{x}}^{{3}}{+}{12}{}{x}\right){}{\mathrm{Dx}}{+}{{x}}^{{4}}{+}{6}{}{{x}}^{{2}}{+}{3}$ (6)

Mixed differential-difference case:

 > $A≔\mathrm{skew_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx},x\right],\mathrm{shift}=\left[\mathrm{Sn},n\right]\right):$
 > $\mathrm{skew_product}\left(\mathrm{Dx},x,A\right),\mathrm{skew_product}\left(\mathrm{Sn},n,A\right)$
 ${\mathrm{Dx}}{}{x}{+}{1}{,}\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (7)
 > $\mathrm{skew_product}\left(\mathrm{Dx}\mathrm{Sn},xn,A\right)$
 $\left({x}{}{n}{+}{x}\right){}{\mathrm{Dx}}{}{\mathrm{Sn}}{+}\left({n}{+}{1}\right){}{\mathrm{Sn}}$ (8)
 > $\mathrm{skew_product}\left(\mathrm{Sn},{n}^{5},A\right)$
 $\left({{n}}^{{5}}{+}{5}{}{{n}}^{{4}}{+}{10}{}{{n}}^{{3}}{+}{10}{}{{n}}^{{2}}{+}{5}{}{n}{+}{1}\right){}{\mathrm{Sn}}$ (9)
 > $\mathrm{skew_product}\left({\mathrm{Sn}}^{5},n,A\right)$
 $\left({n}{+}{5}\right){}{{\mathrm{Sn}}}^{{5}}$ (10)

 See Also