p1 + p2 returns the sum of the terms p1 and p2. The result is a power series.

The calling sequence Add(P) returns the sum of the terms in P.

The calling sequence Add(P, coefficients = C) returns the sum of the products C[i] * P[i]. The length of the list C must be the same as the number of elements of P.

u1 + u2 returns the sum of the terms u1 and u2. The result is a univariate polynomial over power series or over Puiseux series.

The calling sequence Add(U) returns the sum of the entries of 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.

s1 + s2 returns the sum of the terms s1 and s2 as long as their Puiseux series orders are compatible. The result is a Puiseux series.

Two variable orders are compatible if, whenever two variables occur in both orders, they occur in the same order. For example, $\left[x\,y\,z\right]$ and $\left[t\,x\,z\,w\right]$ are compatible, because the only variables that occur in both orders are $x$ and $z$, and $x$ occurs before $z$ in both orders.

The calling sequence Add(S) returns the sum of the entries of S as long as this is possible. They are converted to Puiseux series. If this is not possible, an error is raised.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

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

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

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

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

$a+b$

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

$a+1$

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

$\mathrm{Add}\left(a\,b\,c\,1+xyz\right)$

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

$\mathrm{Add}\left(a\,b\,c\,\mathrm{coefficients}=\left[1\,5\,10\right]\right)$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}+5+5x+5y+5z+30z^3+20xy\mathrm{:}6+6x+6y+5z+\mathrm{\dots }\right]$

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

$f+z+3$

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

$\mathrm{Add}\left(f\,z+3\right)$

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

$f+\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)$

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

$g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\,\mathrm{PowerSeries}\left(3\right)\right]\,z\right)\:$

$f+g$

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

$h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\,\mathrm{PowerSeries}\left(3\right)\right]\,w\right)\:$

$f+h$

Error, (in MultivariatePowerSeries:-Add) incompatible inputs: expect UnivariatePolynomialOverPowerSeries with the same main variable, but received w <> z

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

$d≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}0+\mathrm{\dots }\right]$

$f+d$

Error, (in MultivariatePowerSeries:-Add) 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

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

$e≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}z{\&ExponentialE;}^x\mathrm{:}0+\mathrm{\dots }\right]$

$f+e$

$\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s:}\left(0\right)\mathrm{+}\left(1+\mathrm{\dots }\right)z\mathrm{+}\left(y\right)z^2\mathrm{+}\left(xy\right)z^3\right]$

$\mathrm{s1}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+u}\right)\&comma;\left[u=x^{-\frac{1}{3}}{y}^2\right]\&comma;\left[x=3\&comma;y=-4\right]\right)$

$\mathrm{s1}≔\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s2}≔\mathrm{PuiseuxSeries}\left(2+2\left(u+v\right)\&comma;\left[u=x^{-\frac{1}{2}}y\&comma;v=y\right]\&comma;\left[x=3\&comma;y=2\right]\right)$

$\mathrm{s2}≔\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\left(2+\frac{2y}{\sqrt{x}}+2y\right)x^3{y}^2\mathrm{:}2x^3{y}^2+2x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{y}^3+2y^3{x}^3\right]$

$\mathrm{s3}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+uv}\right)\&comma;\left[y\&comma;x\right]\&comma;\left[u\&comma;v\right]\&comma;\left[\left[1\&comma;0\right]\&comma;\left[1\&comma;-\frac{1}{2}\right]\right]\right)$

$\mathrm{s3}≔\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{1}{\frac{y^2}{\sqrt{x}}+1}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{s1}+\mathrm{s2}$

$\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{\left(\frac{1}{1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}+\left(2+\mathrm{\dots }++2y\right)y^6\right)x^3}{y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}+1+xy$

$\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{\left(\frac{1}{1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}+\frac{\left(xy+1\right)y^4}{x^3}\right)x^3}{y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}+a$

$\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{\left(\frac{1}{1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}+\frac{y^4}{\left(1-x-y\right)x^3}\right)x^3}{y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}+f$

$\left[\mathrm{Puis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{\left(\frac{1}{1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}+\frac{\left(\mathrm{\dots }\mathrm{\dots }+\mathrm{\dots }\right)y^4}{x^3}\right)x^3}{y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}+\mathrm{s3}$

Error, the order of Puiseux series [x, y] and [y, x] are not compatible

$\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s1}\right)$

$\left[x\&comma;y\right]$

$\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s3}\right)$

$\left[y\&comma;x\right]$

$u≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right)\&comma;\mathrm{PuiseuxSeries}\left(0\right)\&comma;\mathrm{PuiseuxSeries}\left(x\&comma;\left[x=x^{\frac{1}{3}}\right]\right)\&comma;\mathrm{PuiseuxSeries}\left(y\&comma;\left[y=y^{\frac{1}{2}}\right]\right)\&comma;\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y}\&comma;\left[x=x{y}^{\frac{1}{2}}\&comma;y=x{y}^{-1}\right]\right)\right]\&comma;z\right)$

$u≔\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPuis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)z^2\mathrm{+}\left(\sqrt{y}\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

$u+f$

$\left[\mathrm{Univariat\&ExponentialE;PolynomialOv\&ExponentialE;rPuis\&ExponentialE;uxS\&ExponentialE;ri\&ExponentialE;s:}\left(1\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

Monforte, A.A., & Kauers, M. "Formal Laurent series in several variables." Expositiones Mathematicae. Vol. 31 No. 4 (2013): 350-367.

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

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

The MultivariatePowerSeries[Add] command was updated in Maple 2023.

The s1, s2 and S parameters were introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.