Coefficient - Maple Help

OreTools[Utility]

 Coefficient
 return the coefficient of a specific power in an Ore polynomial
 Coefficients
 return the coefficient sequence of an Ore polynomial
 Degree
 return the degree of an Ore polynomial with respect to the noncommutative indeterminate
 return the leading coefficient of an Ore polynomial
 LowDegree
 return the degree of the least power with nonzero coefficient
 RandOrePoly
 return a random Ore polynomial
 TrailingCoefficient
 return the trailing coefficient of an Ore Polynomial
 VariableDegree
 return the maximal degree of the coefficients of an Ore Polynomial in the variable in an Ore algebra

 Calling Sequence Coefficient(Poly, n) Coefficients(Poly) Degree(Poly) LeadingCoefficient(Poly) LowDegree(Poly) RandOrePoly(A, opts) TrailingCoefficient(Poly) VariableDegree(Poly, A)

Parameters

 Poly - Ore polynomial; to define an Ore polynomial, see OreTools/OrePoly n - non-negative integer A - Ore algebra opts - options

Description

 • The Coefficient(Poly, n) calling sequence returns the coefficient of the nth power of the noncommutative indeterminate in Poly.
 • The Coefficients(Poly) calling sequence returns the sequence of coefficients of Poly.
 • The Degree(Poly) calling sequence returns the degree of Poly with respect to the noncommutative indeterminate.
 • The LowDegree(Poly) calling sequence returns the trailing degree of Poly.
 • The RandOrePoly(A) calling sequence returns a random Ore polynomial in the Ore algebra A.
 The first argument A specifies the ring in which the polynomial is to be generated. The possible options are:
 coeffs - Generate the coefficients
 terms - Number of terms in the noncommutative indeterminate
 degree - Degree on the noncommutative indeterminate
 • The TrailingCoefficient(Poly) calling sequence returns the trailing coefficient of A.
 • The VariableDegree(Poly, A) calling sequence returns the maximal degree of coefficients of Poly with respect to the variable in A.  Note that the coefficients of Poly are supposed to be polynomials in the variable.
 • For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

Examples

 > $\mathrm{with}\left(\mathrm{OreTools}\right):$
 > $\mathrm{with}\left(\mathrm{OreTools}\left[\mathrm{Utility}\right]\right):$
 > $\mathrm{Poly}≔'\mathrm{OrePoly}'\left(0,2n-1,0,\frac{1}{n}\right)$
 ${\mathrm{Poly}}{≔}{\mathrm{OrePoly}}{}\left({0}{,}{2}{}{n}{-}{1}{,}{0}{,}\frac{{1}}{{n}}\right)$ (1)
 > $\mathrm{Coefficient}\left(\mathrm{Poly},1\right)$
 ${2}{}{n}{-}{1}$ (2)
 > $\mathrm{Coefficient}\left(\mathrm{Poly},6\right)$
 ${0}$ (3)
 > $\mathrm{Coefficients}\left(\mathrm{Poly}\right)$
 ${0}{,}{2}{}{n}{-}{1}{,}{0}{,}\frac{{1}}{{n}}$ (4)
 > $\mathrm{Degree}\left(\mathrm{Poly}\right)$
 ${3}$ (5)
 > $\mathrm{Degree}\left('\mathrm{OrePoly}'\left(0\right)\right)$
 ${-}{\mathrm{\infty }}$ (6)
 > $\mathrm{LeadingCoefficient}\left(\mathrm{Poly}\right)$
 $\frac{{1}}{{n}}$ (7)
 > $\mathrm{LowDegree}\left(\mathrm{Poly}\right)$
 ${1}$ (8)
 > $\mathrm{LowDegree}\left('\mathrm{OrePoly}'\left(0\right)\right)$
 ${\mathrm{\infty }}$ (9)
 > $\mathrm{TrailingCoefficient}\left(\mathrm{Poly}\right)$
 ${2}{}{n}{-}{1}$ (10)
 > $\mathrm{TrailingCoefficient}\left('\mathrm{OrePoly}'\left(0\right)\right)$
 ${0}$ (11)
 > $A≔\mathrm{SetOreRing}\left(n,'\mathrm{shift}'\right):$
 > $\mathrm{Poly}≔\mathrm{RandOrePoly}\left(A\right)$
 ${\mathrm{Poly}}{≔}{\mathrm{OrePoly}}{}\left({72}{}{{n}}^{{5}}{+}{37}{}{{n}}^{{4}}{-}{23}{}{{n}}^{{3}}{+}{87}{}{{n}}^{{2}}{+}{44}{}{n}{+}{29}{,}{-}{50}{}{{n}}^{{5}}{+}{23}{}{{n}}^{{4}}{+}{75}{}{{n}}^{{3}}{-}{92}{}{{n}}^{{2}}{+}{6}{}{n}{+}{74}{,}{-}{17}{}{{n}}^{{5}}{-}{75}{}{{n}}^{{4}}{-}{10}{}{{n}}^{{3}}{-}{7}{}{{n}}^{{2}}{-}{40}{}{n}{+}{42}{,}{-}{10}{}{{n}}^{{5}}{+}{62}{}{{n}}^{{4}}{-}{82}{}{{n}}^{{3}}{+}{80}{}{{n}}^{{2}}{-}{44}{}{n}{+}{71}{,}{-}{62}{}{{n}}^{{4}}{+}{97}{}{{n}}^{{3}}{-}{73}{}{{n}}^{{2}}{-}{4}{}{n}{-}{83}{,}{-}{7}{}{{n}}^{{5}}{+}{22}{}{{n}}^{{4}}{-}{55}{}{{n}}^{{3}}{-}{94}{}{{n}}^{{2}}{+}{87}{}{n}{-}{56}\right)$ (12)
 > $\mathrm{Degree}\left(\mathrm{Poly}\right)$
 ${5}$ (13)
 > $\mathrm{VariableDegree}\left(\mathrm{Poly},A\right)$
 ${5}$ (14)
 > $\mathrm{VariableDegree}\left('\mathrm{OrePoly}'\left(0\right),A\right)$
 ${-}{\mathrm{\infty }}$ (15)
 > $B≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right):$
 > $C≔\mathrm{RandOrePoly}\left(B,\mathrm{coeffs}=\mathrm{polynom}\left(\mathrm{degree}=3,\mathrm{terms}=2\right),\mathrm{terms}=2,\mathrm{degree}=10\right)$
 ${C}{≔}{\mathrm{OrePoly}}{}\left({0}{,}{0}{,}{0}{,}{0}{,}{40}{}{{x}}^{{3}}{-}{81}{}{x}{,}{11}{+}{95}{}{x}\right)$ (16)
 > $\mathrm{VariableDegree}\left(C,B\right)$
 ${3}$ (17)
 > $F≔\mathrm{SetOreRing}\left(\left[x,q\right],'\mathrm{qshift}'\right):$
 > $\mathrm{Utility}\left[\mathrm{RandOrePoly}\right]\left(F,\mathrm{coeffs}=\mathrm{ratpoly}\left(\mathrm{degnum}=1,\mathrm{degden}=2,\mathrm{terms}=2,\mathrm{degree}=5\right),\mathrm{terms}=3,\mathrm{degree}=10\right)$
 ${\mathrm{OrePoly}}{}\left({0}{,}\frac{{-}{87}{}{q}{+}{47}{}{x}{-}{90}}{{-}{88}{-}{48}{}{q}}{,}{0}{,}\frac{{16}{}{q}{+}{30}{}{x}{-}{27}}{{72}{}{q}{}{x}{-}{96}}{,}{0}{,}{0}{,}{0}{,}{0}{,}\frac{{-}{51}{}{q}{+}{77}{}{x}{+}{95}}{{-}{28}{}{q}{}{x}{+}{55}{}{q}}\right)$ (18)