MultivariatePowerSeries/SumOfAllMonomials - Maple Help

MultivariatePowerSeries

 SumOfAllMonomials
 Get the power series representing the sum of all monomials over a set of variables

 Calling Sequence SumOfAllMonomials(x)

Parameters

 x - list of variables

Description

 • The command SumOfAllMonomials(x) returns the power series containing each monomial in the variables x with coefficient 1; that is, its homogeneous part of degree d is the sum of all monomials of degree d.
 • 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

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

We create a power series that is the sum of all monomials in $x$, and compute its homogeneous part of degree 2 and its truncation at precision 4.

 > $a≔\mathrm{SumOfAllMonomials}\left(\left[x\right]\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}}{:}{1}{+}{x}{+}{\dots }\right]$ (1)
 > $\mathrm{HomogeneousPart}\left(a,2\right)$
 ${{x}}^{{2}}$ (2)
 > $\mathrm{Truncate}\left(a,4\right)$
 ${{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{x}{+}{1}$ (3)

We create a power series that is the sum of all monomials in $x$, $y$, and/or $z$. We compute its homogeneous part of degree 2 and its truncation at precision 3.

 > $b≔\mathrm{SumOfAllMonomials}\left(\left[x,y,z\right]\right)$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{\left({1}{-}{x}\right){}\left({1}{-}{y}\right){}\left({1}{-}{z}\right)}{:}{1}{+}{x}{+}{y}{+}{z}{+}{\dots }\right]$ (4)
 > $\mathrm{HomogeneousPart}\left(b,2\right)$
 ${{x}}^{{2}}{+}{y}{}{x}{+}{z}{}{x}{+}{{y}}^{{2}}{+}{z}{}{y}{+}{{z}}^{{2}}$ (5)
 > $\mathrm{Truncate}\left(b,3\right)$
 ${{x}}^{{3}}{+}{{x}}^{{2}}{}{y}{+}{{x}}^{{2}}{}{z}{+}{x}{}{{y}}^{{2}}{+}{x}{}{y}{}{z}{+}{x}{}{{z}}^{{2}}{+}{{y}}^{{3}}{+}{{y}}^{{2}}{}{z}{+}{y}{}{{z}}^{{2}}{+}{{z}}^{{3}}{+}{{x}}^{{2}}{+}{y}{}{x}{+}{z}{}{x}{+}{{y}}^{{2}}{+}{z}{}{y}{+}{{z}}^{{2}}{+}{x}{+}{y}{+}{z}{+}{1}$ (6)

Compatibility

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