MultivariatePowerSeries/UnivariatePolynomialOverPowerSeries - Maple Help

MultivariatePowerSeries

 UnivariatePolynomialOverPowerSeries
 Create a univariate polynomial over power series

 Calling Sequence UnivariatePolynomialOverPowerSeries(lp,v) UnivariatePolynomialOverPowerSeries(p,v) UnivariatePolynomialOverPowerSeries(ps, v) UnivariatePolynomialOverPowerSeries(u) UnivariatePolynomialOverPowerSeries(u, v)

Parameters

 lp - list, Array, or Vector of power series generated by this package v - variable p - polynomial ps - power series generated by this package u - univariate polynomial over power series generated by this package

Description

 • The function call UnivariatePolynomialOverPowerSeries(lp,v) creates a univariate polynomial over power series with main variable v and with coefficients that are power series from lp. The degree of the resulting polynomial is equal to the length of lp minus one. The coefficient of v^(i-1) is the i-th element of lp. In particular, the first element of lp is the constant coefficient. The main variable, v, cannot occur in any of the power series in lp.
 • The function call UnivariatePolynomialOverPowerSeries(p,v) creates a univariate polynomial with power series coefficients. It returns the same result as UnivariatePolynomialOverPowerSeries(lp,v) where lp := [seq(PowerSeries(coeff(p,v,i)),i=0..degree(p,v))].
 • The function call UnivariatePolynomialOverPowerSeries(ps, v) creates a univariate polynomial over power series representing ps, with v as its main variable. This is only possible if ps is known to be a polynomial function of v, which is the case if it is independent of v (in which case it is trivially polynomial) or if the analytic expression for ps is known and it is polynomial in v. If neither of the former two cases is true, then an error is raised.
 • The function call UnivariatePolynomialOverPowerSeries(u, v) copies u. You may omit v in this case, but if you specify it, it must be equal to the main variable of u.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

Examples

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

Create a univariate polynomial over power series from a list of power series.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right),\mathrm{PowerSeries}\left(0\right),\mathrm{PowerSeries}\left(x\right),\mathrm{PowerSeries}\left(y\right),\frac{1}{\mathrm{PowerSeries}\left(1+x+y\right)}\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({x}\right){}{{z}}^{{2}}{+}\left({y}\right){}{{z}}^{{3}}{+}\left({1}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (1)
 > $\mathrm{Degree}\left(f\right)$
 ${4}$ (2)
 > $\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(f,i\right),i=0..4\right)\right]$
 $\left[\left[{PowⅇrSⅇriⅇs:}{1}\right]{,}\left[{PowⅇrSⅇriⅇs:}{0}\right]{,}\left[{PowⅇrSⅇriⅇs:}{x}\right]{,}\left[{PowⅇrSⅇriⅇs:}{y}\right]{,}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{+}{x}{+}{y}}{:}{1}{+}{\dots }\right]\right]$ (3)

We compute its linear and quadratic truncation. These are defined in terms of the homogeneous degree of the coefficients, ignoring the degree in the main variable.

 > $\mathrm{Truncate}\left(f,1\right)$
 ${1}{+}\left({1}{-}{x}{-}{y}\right){}{{z}}^{{4}}{+}{y}{}{{z}}^{{3}}{+}{x}{}{{z}}^{{2}}$ (4)
 > $\mathrm{Truncate}\left(f,2\right)$
 ${1}{+}\left({{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{-}{x}{-}{y}{+}{1}\right){}{{z}}^{{4}}{+}{y}{}{{z}}^{{3}}{+}{x}{}{{z}}^{{2}}$ (5)

Create a univariate polynomial over power series from a polynomial

 > $p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right)$
 ${p}{≔}\left({z}{-}{1}\right){}\left({z}{-}{2}\right){}\left({z}{-}{3}\right){+}{x}{}\left({{z}}^{{2}}{+}{z}\right)$ (6)
 > $\mathrm{UnivariatePolynomialOverPowerSeries}\left(p,z\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-6}\right){+}\left({11}{+}{x}\right){}{z}{+}\left({-6}{+}{x}\right){}{{z}}^{{2}}{+}\left({1}\right){}{{z}}^{{3}}\right]$ (7)

The following attempt will not work, because Maple cannot determine that d is polynomial in z (though actually it is).

 > $d≔\mathrm{PowerSeries}\left(d↦\mathrm{ifelse}\left(d=0,0,\frac{z\cdot {x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{variables}=\left\{x,z\right\}\right)$
 ${d}{≔}\left[{PowⅇrSⅇriⅇs:}{0}{+}{\dots }\right]$ (8)
 > $\mathrm{UnivariatePolynomialOverPowerSeries}\left(d,z\right)$

We define e in the same way as d but specify the analytic expression. Then we can successfully convert it to a univariate polynomial over power series.

 > $e≔\mathrm{PowerSeries}\left(d↦\mathrm{ifelse}\left(d=0,0,\frac{z\cdot {x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{analytic}=z\mathrm{exp}\left(x\right)\right)$
 ${e}{≔}\left[{PowⅇrSⅇriⅇs of}{z}{}{{ⅇ}}^{{x}}{:}{0}{+}{\dots }\right]$ (9)
 > $k≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(e,z\right)$
 ${k}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (10)

To copy k, we can specify the main variable explicitly or omit it. If we specify it explicitly, it has to be z, otherwise we obtain an error:

 > $\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k,x\right)$
 > $\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k,z\right)$
 ${\mathrm{k1}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (11)
 > $\mathrm{k2}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\right)$
 ${\mathrm{k2}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (12)

Compatibility

 • The MultivariatePowerSeries[UnivariatePolynomialOverPowerSeries] command was introduced in Maple 2021.