The command p1 * p2 returns the product of p1 and p2. The result is a power series.

The command Multiply(P) returns the product of the factors in P.

The command u1 * u2 returns the product of u1 and u2. The result is a univariate polynomial over power series.

The command Multiply(U) returns the product of the factors in U. They are converted to univariate polynomials over power series in the same variable. If this is not possible, an error is raised. This may happen if there are univariate polynomials over power series in different variables. It can also happen if the univariate polynomials over power series all have the same main variable, say x, but one of the other arguments is a power series that is not known to be expressible as a polynomial in x. The same restrictions apply to the calling sequence u1 * u2.

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.

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

$a≔\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\:$

$b≔\mathrm{PowerSeries}\left(1+x+y+z\right)\:$

$ab$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1+x+y+z}{1-x-y}\mathrm{:}1+2x+2y+z+\mathrm{\dots }\right]$

$a\left(x+y\right)$

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

$c≔\mathrm{PowerSeries}\left(2xy+3z^3\right)\:$

$d≔\mathrm{Multiply}\left(a\,b\,c\,1+x+y\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{\left(1+x+y+z\right)\left(3z^3+2xy\right)\left(1+x+y\right)}{1-x-y}\mathrm{:}0+\mathrm{\dots }\right]$

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

$2xy$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(xz+y{z}^2+xy{z}^3\,z\right)\:$

$fb$

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

$g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\right]\,z\right)\:$

$h≔fg$

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

$k≔\mathrm{Multiply}\left(f\,g\right)$

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

$\mathrm{Truncate}\left(h-k\,10\right)$

$0$

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

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