The command TaylorShift(u, c) returns the univariate polynomial over power series obtained by substituting v + c for v in u, where v is the main variable of u. In other words, the result is obtained by composing u with the map that sends v to v + c.

A typical usage is when c is a root of the polynomial returned by EvaluateAtOrigin(u). This happens, for example, in HenselFactorize.

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.

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

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

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

$\mathrm{f1}≔\mathrm{TaylorShift}\left(f\,1\right)$

$\mathrm{f1}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(3+\mathrm{\dots }\right)\mathrm{+}\left(3+x+3y+\mathrm{\dots }\right)z\mathrm{+}\left(1+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{f0}≔\mathrm{TaylorShift}\left(\mathrm{f1}\,-1\right)$

$\mathrm{f0}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(1+x+y+\mathrm{\dots }\right)z\mathrm{+}\left(1+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{ApproximatelyEqual}\left(f\,\mathrm{f0}\,20\right)$

$\mathrm{true}$

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

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