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.

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

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right)\,\mathrm{PowerSeries}\left(0\right)\,\mathrm{PowerSeries}\left(x\right)\,\mathrm{PowerSeries}\left(y\right)\,\frac{1}{\mathrm{PowerSeries}\left(1+x+y\right)}\right]\,z\right)$

$f≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x\right)z^2\mathrm{+}\left(y\right)z^3\mathrm{+}\left(1+\mathrm{\dots }\right)z^4\right]$

$\mathrm{Degree}\left(f\right)$

$4$

$\left[\mathrm{seq}\left(\mathrm{GetCoefficient}\left(f\,i\right)\,i=0..4\right)\right]$

$\left[\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}x\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}y\right]\,\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1+x+y}\mathrm{:}1+\mathrm{\dots }\right]\right]$

$\mathrm{Truncate}\left(f\,1\right)$

$1+\left(1-x-y\right)z^4+y{z}^3+x{z}^2$

$\mathrm{Truncate}\left(f\,2\right)$

$1+\left(x^2+2xy+y^2-x-y+1\right)z^4+y{z}^3+x{z}^2$

$p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)$

$p≔\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)$

$\mathrm{UnivariatePolynomialOverPowerSeries}\left(p\,z\right)$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(\mathrm{-6}\right)\mathrm{+}\left(11+x\right)z\mathrm{+}\left(\mathrm{-6}+x\right)z^2\mathrm{+}\left(1\right)z^3\right]$

$d≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\,0\,\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\,\mathrm{variables}=\left\{x\,z\right\}\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}0+\mathrm{\dots }\right]$

$\mathrm{UnivariatePolynomialOverPowerSeries}\left(d\,z\right)$

Error, (in MultivariatePowerSeries:-UnivariatePolynomialOverPowerSeries) attempted to convert a power series involving z to a univariate polynomial over power series in z, but it is not known to be polynomial in z

$e≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d=0\,0\,\frac{z\cdot x^{d-1}}{\left(d-1\right)!}\right)\,\mathrm{analytic}=z\exp \left(x\right)\right)$

$e≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}z{\ⅇ}^x\mathrm{:}0+\mathrm{\dots }\right]$

$k≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(e\,z\right)$

$k≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

$\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\,x\right)$

Error, (in MultivariatePowerSeries:-UnivariatePolynomialOverPowerSeries) you specified x as the main variable, but the main variable of the first argument is z

$\mathrm{k1}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\,z\right)$

$\mathrm{k1}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

$\mathrm{k2}≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(k\right)$

$\mathrm{k2}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\right]$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.