We define a power series for .
We shift x by 1 and y by 2, performing all steps three times: all at once, x first, or y first. (In practice, one would typically do both shifts at once: it is computationally more efficient.)
Let's take a look at the first few homogeneous components of ps_both.
Now we verify that the three results, ps_both, ps_x_y and ps_y_x, are equal (up to homogeneous degree 20).
We define a univariate polynomial over power series.
We apply a Taylor shift by 1, and then by -1 on the result.
We verify that the result is equal to the original polynomial (up to homogeneous degree 20).