Formal Power Series - Maple Help

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Formal Power Series

The convert/FormalPowerSeries functionality was completely rewritten for Maple 2022. It offers a number of advantages over previous versions:

 • Closed-form solutions can be found in a number of cases where previous versions failed.
 • Solutions in terms of $m$-fold hypergeometric sequences for arbitrary positive integers $m$ are now supported in more cases than before.
 • Notwithstanding the name, formal Laurent and Puiseux series (i.e., with negative or fractional exponents) can be computed as well, now in more cases than before.
 • convert/FormalPowerSeries will automatically attempt to return the series coefficients in purely real form, making the previous option makereal obsolete.
 • In a number of cases, the new code returns more compact answers than previous versions.
 • If a closed form expression for the power series coefficients cannot be found, and a recurrence relation of degree 1 or 2 exists, it will be returned instead. Previously, only linear recurrences could be computed, and would only be returned if option recurrence was specified.
 • When a recurrence relation is returned, now the initial conditions are given as well.
 • Additional options give more control over the underlying algorithm(s) used and the form of the output.

Maple 2021

Maple 2022

More closed-form solutions, notably, for sums of several terms and Puiseux solutions.

 $\left({-}\frac{{1}}{{2}}{}{z}{+}\frac{{1}}{{6}}{}{{z}}^{{3}}\right){}{\mathrm{arctan}}{}\left({z}\right)$ (1)

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{k}}{}{{z}}^{{2}{}{k}{+}{2}}}{{4}{}{k}{+}{2}}\right){+}\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{k}}{}{{z}}^{{2}{}{k}{+}{4}}}{{12}{}{k}{+}{6}}\right)$ (2)

 ${-}\frac{{{z}}^{{2}}}{{6}}{+}\frac{{5}}{{9}}{+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({4}{}{n}{-}{5}\right){}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}}}{{3}{}\left({2}{}{n}{-}{1}\right){}\left({2}{}{n}{-}{3}\right)}\right)$ (3)

$\mathrm{convert}\left(\mathrm{arctan}\left(z\right)+\mathrm{arcsin}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 ${\mathrm{arctan}}{}\left({z}\right){+}{\mathrm{arcsin}}{}\left({z}\right)$ (4)

$\mathrm{map}\left(\mathrm{convert},\mathrm{arctan}\left(z\right)+\mathrm{arcsin}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 $\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{k}}{}{{z}}^{{2}{}{k}{+}{1}}}{{2}{}{k}{+}{1}}\right){+}\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({2}{}{k}\right){!}{}{{4}}^{{-}{k}}{}{{z}}^{{2}{}{k}{+}{1}}}{{{k}{!}}^{{2}}{}\left({2}{}{k}{+}{1}\right)}\right)$ (5)

$\mathrm{convert}\left(\mathrm{arctan}\left(z\right)+\mathrm{arcsin}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\left({-1}\right)}^{{n}}{}{{n}{!}}^{{2}}{+}\left({2}{}{n}\right){!}{}{{4}}^{{-}{n}}\right){}{{z}}^{{2}{}{n}{+}{1}}}{\left({2}{}{n}{+}{1}\right){}{{n}{!}}^{{2}}}$ (6)

$\mathrm{convert}\left(\mathrm{arctan}\left(z\right)+\mathrm{arcsin}\left(z\right),\mathrm{FormalPowerSeries},\mathrm{output}=\mathrm{expanded}\right)$

 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}{n}{+}{1}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({2}{}{n}\right){!}{}{{4}}^{{-}{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{\left({2}{}{n}{+}{1}\right){}{{n}{!}}^{{2}}}\right)$ (7)

 $\frac{{\left({1}{-}\sqrt{{1}{-}{4}{}{z}}\right)}^{{2}}{}{{z}}^{{2}}}{{4}{}\sqrt{{1}{-}{4}{}{z}}}$ (8)

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({2}{}{n}{+}{2}\right){!}{}\left({n}{+}{2}\right){}\left({n}{+}{1}\right){}{{z}}^{{n}{+}{4}}}{{\left({n}{+}{2}\right){!}}^{{2}}}$ (9)

 $\sqrt{\sqrt{{8}{}{{z}}^{{3}}{+}{1}}{-}{1}}$ (10)

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{2}}^{{-}{n}{+}{1}}{}\left({2}{}{n}{+}{1}\right){}\left({4}{}{n}\right){!}{}{{z}}^{{3}{}{n}{+}\frac{{3}}{{2}}}}{{\left({2}{}{n}{+}{1}\right){!}}^{{2}}}$ (11)

Solutions in purely real form by default.

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{I}{}{{I}}^{{k}}{}\left({k}{+}{1}\right)}{{2}{}{k}{!}}{+}\frac{{I}{}{\left({-I}\right)}^{{k}}{}\left({k}{+}{1}\right)}{{2}{}{k}{!}}\right){}{{z}}^{{k}}$ (12)

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{sin}}{}\left(\frac{{k}{}{\mathrm{\pi }}}{{2}}\right){}\left({k}{+}{1}\right){}{{z}}^{{k}}}{{k}{!}}$ (13)

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{\left({-1}\right)}^{{n}}{}\left({n}{+}{1}\right){}{{z}}^{{2}{}{n}{+}{1}}}{\left({2}{}{n}{+}{1}\right){!}}$ (14)

$\mathrm{convert}\left(\mathrm{ln}\left(1+z+{z}^{2}+{z}^{3}\right),\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{k}{+}{1}}}{{k}{+}{1}}{-}\frac{{{I}}^{{k}{+}{1}}}{{k}{+}{1}}{-}\frac{{\left({-I}\right)}^{{k}{+}{1}}}{{k}{+}{1}}\right){}{{z}}^{{k}{+}{1}}$ (15)

$\mathrm{convert}\left(\mathrm{ln}\left(1+z+{z}^{2}+{z}^{3}\right),\mathrm{FormalPowerSeries},\mathrm{makereal}\right)$

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\left({-1}\right)}^{{k}}{+}{2}{}{\mathrm{sin}}{}\left(\frac{{k}{}{\mathrm{\pi }}}{{2}}\right)\right){}{{z}}^{{k}{+}{1}}}{{k}{+}{1}}$ (16)

$\mathrm{convert}\left(\mathrm{ln}\left(1+z+{z}^{2}+{z}^{3}\right),\mathrm{FormalPowerSeries}\right)$

 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{{n}{+}{1}}}{{n}{+}{1}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}{+}{2}}}{{n}{+}{1}}\right)$ (17)

 $\frac{{{z}}^{{2}}}{{3}}{+}\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}\sqrt{{3}}{}\left({\left({-}\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left({-}\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{}{\left(\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{-}{\left({-}\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left({-}\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{}{\left(\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{+}{\left({-}\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left(\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left(\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{-}{\left({-}\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{}{\left(\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left(\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}\right){}{{z}}^{{k}}}{{9}{}{\left({-}\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left({-}\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}{}{\left(\frac{{I}}{{3}}{}{{3}}^{{3}}{{4}}}\right)}^{{k}}{}{\left(\frac{{{3}}^{{3}}{{4}}}}{{3}}\right)}^{{k}}}\right)$ (18)

 $\frac{{{z}}^{{2}}}{{3}}{+}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{2}{}{{z}}^{{k}}{}{{3}}^{\frac{{k}}{{4}}{+}\frac{{1}}{{2}}}{}\left({2}{}{\mathrm{cos}}{}\left(\frac{{k}{}{\mathrm{\pi }}}{{2}}\right){-}{\left({-1}\right)}^{{k}}{-}{1}\right)}{{9}}\right)$ (19)

 $\frac{{{z}}^{{2}}}{{3}}{+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{8}{}{{3}}^{{n}{-}{1}}{}{{z}}^{{4}{}{n}{+}{2}}\right)$ (20)

 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{3}}{+}{2}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{4}}{{3}}}{}{\left({-1}\right)}^{{2}}{{3}}}{}\mathrm{q__1}{-}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{+}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__1}}^{{3}}{+}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{-}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{-}{2}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{2}}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\mathrm{q__2}}^{{4}}{{3}}}{}\mathrm{q__1}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{4}}{{3}}}{}\mathrm{q__1}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{4}}{{3}}}{}{\left({-1}\right)}^{{2}}{{3}}}{}\mathrm{q__1}{+}{4}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{2}}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__1}}^{{2}}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__1}}^{{2}}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{2}}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{2}}{-}{2}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{{3}}}{}{\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__1}}^{{2}}{+}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{}{\left({-1}\right)}^{{2}}{{3}}}{+}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{}{\left({-1}\right)}^{{2}}{{3}}}{-}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{}{\left({-1}\right)}^{{1}}{{3}}}{-}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__2}}^{{2}}{}{\left({-1}\right)}^{{1}}{{3}}}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{-}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__1}}^{{3}}{{2}}}{}\mathrm{q__2}{+}{2}{}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\mathrm{q__1}}^{{3}}{+}{2}{}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\mathrm{q__1}}^{{3}}\right){}{{z}}^{{k}}}{{2}{}{\left({\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}{\left({-}\sqrt{\mathrm{q__1}}\right)}^{{k}}{}\left(\sqrt{\mathrm{q__1}}{+}{\mathrm{q__2}}^{{1}}{{3}}}\right){}{\left(\sqrt{\mathrm{q__1}}\right)}^{{k}}{}\mathrm{q__1}{}\left({-}\sqrt{\mathrm{q__1}}{+}{\mathrm{q__2}}^{{1}}{{3}}}\right){}{\left({-}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}\left({-}\sqrt{\mathrm{q__1}}{+}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right){}\left(\sqrt{\mathrm{q__1}}{+}{\left({-1}\right)}^{{1}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right){}{\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}\right)}^{{k}}{}\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}{+}\sqrt{\mathrm{q__1}}\right){}\left({\left({-1}\right)}^{{2}}{{3}}}{}{\mathrm{q__2}}^{{1}}{{3}}}{-}\sqrt{\mathrm{q__1}}\right){}{\left({\left({-1}\right)}^{{1}}{{3}}}{+}{1}\right)}^{{2}}{}\left({\left({-1}\right)}^{{1}}{{3}}}{-}{1}\right){}\mathrm{q__2}}$ (21)

$\mathrm{convert}\left(\frac{1}{\left(\mathrm{q__1}-{z}^{2}\right)\cdot \left(\mathrm{q__2}-{z}^{3}\right)},\mathrm{FormalPowerSeries},z\right)$

 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({-}{\mathrm{q__1}}^{{2}{-}\frac{{n}}{{2}}}{+}{\left({-1}\right)}^{{n}}{}{\mathrm{q__1}}^{{2}{-}\frac{{n}}{{2}}}{+}{2}{}{\mathrm{q__2}}^{{-}\frac{{1}}{{3}}{-}\frac{{n}}{{3}}}{}{\mathrm{q__1}}^{{5}}{{2}}}{-}{2}{}{\mathrm{q__1}}^{{-}\frac{{n}}{{2}}{+}\frac{{1}}{{2}}}{}\mathrm{q__2}{-}{\left({-1}\right)}^{{n}}{}{\mathrm{q__1}}^{{-}{1}{-}\frac{{n}}{{2}}}{}{\mathrm{q__2}}^{{2}}{+}{2}{}{\mathrm{q__2}}^{\frac{{2}}{{3}}{-}\frac{{n}}{{3}}}{}\mathrm{q__1}{-}{\mathrm{q__1}}^{{-}{1}{-}\frac{{n}}{{2}}}{}{\mathrm{q__2}}^{{2}}\right){}{{z}}^{{n}}}{{2}{}\left({\mathrm{q__1}}^{{3}}{-}{\mathrm{q__2}}^{{2}}\right){}\left({\mathrm{q__1}}^{{3}}{{2}}}{+}\mathrm{q__2}\right)}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\mathrm{q__1}{}{\mathrm{q__2}}^{{-}{n}{-}{1}}{}{{z}}^{{3}{}{n}}}{{\mathrm{q__2}}^{{4}}{{3}}}{+}{\mathrm{q__2}}^{{2}}{{3}}}{}\mathrm{q__1}{+}{\mathrm{q__1}}^{{2}}}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\mathrm{q__2}}^{{-}{n}{-}\frac{{2}}{{3}}}{}{{z}}^{{3}{}{n}{+}{1}}}{{\mathrm{q__2}}^{{4}}{{3}}}{+}{\mathrm{q__2}}^{{2}}{{3}}}{}\mathrm{q__1}{+}{\mathrm{q__1}}^{{2}}}\right)$ (22)

Recurrence relations returned automatically if no closed form can be found, with initial conditions.
Non-linear (degree 2) recurrences can be computed.

 ${{\mathrm{arcsin}}{}\left({z}\right)}^{{3}}$ (23)

 ${{k}}^{{4}}{}{a}{}\left({k}\right){-}{2}{}\left({k}{+}{1}\right){}\left({k}{+}{2}\right){}\left({{k}}^{{2}}{+}{2}{}{k}{+}{2}\right){}{a}{}\left({k}{+}{2}\right){+}\left({k}{+}{1}\right){}\left({k}{+}{2}\right){}\left({k}{+}{3}\right){}\left({k}{+}{4}\right){}{a}{}\left({k}{+}{4}\right){=}{0}$ (24)

$\mathrm{convert}\left(\mathrm{arcsin}{\left(z\right)}^{3},\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}{+}{1}}{,}{\mathrm{RESol}}{}\left(\left\{\left({{n}}^{{4}}{+}{4}{}{{n}}^{{3}}{+}{6}{}{{n}}^{{2}}{+}{4}{}{n}{+}{1}\right){}{A}{}\left({n}\right){+}\left({-}{2}{}{{n}}^{{4}}{-}{18}{}{{n}}^{{3}}{-}{62}{}{{n}}^{{2}}{-}{98}{}{n}{-}{60}\right){}{A}{}\left({n}{+}{2}\right){+}\left({{n}}^{{4}}{+}{14}{}{{n}}^{{3}}{+}{71}{}{{n}}^{{2}}{+}{154}{}{n}{+}{120}\right){}{A}{}\left({n}{+}{4}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{0}{,}{A}{}\left({2}\right){=}{1}{,}{A}{}\left({3}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (25)

 $\frac{{z}}{{{ⅇ}}^{{z}}{-}{1}}$ (26)

 $\frac{{z}}{{{ⅇ}}^{{z}}{-}{1}}$ (27)

$\mathrm{convert}\left(\frac{z}{{ⅇ}^{z}-1},\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{3}\right){+}\frac{{A}{}\left({n}{+}{2}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{2}}{}{A}{}\left({\mathrm{_k}}\right){}{A}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right)}{{n}{+}{4}}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{1}{,}{A}{}\left({1}\right){=}{-}\frac{{1}}{{2}}{,}{A}{}\left({2}\right){=}\frac{{1}}{{12}}\right\}{,}{\mathrm{INFO}}\right)$ (28)

$\mathrm{convert}\left(\mathrm{LambertW}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 ${\mathrm{LambertW}}{}\left({z}\right)$ (29)

$\mathrm{convert}\left(\mathrm{LambertW}\left(z\right),\mathrm{FormalPowerSeries},\mathrm{recurrence}\right)$

 ${\mathrm{LambertW}}{}\left({z}\right)$ (30)

$\mathrm{convert}\left(\mathrm{LambertW}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{4}\right){+}\frac{{A}{}\left({n}{+}{3}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{2}}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right)}{{n}{+}{3}}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{1}{,}{A}{}\left({2}\right){=}{-1}{,}{A}{}\left({3}\right){=}\frac{{3}}{{2}}\right\}{,}{\mathrm{INFO}}\right)$ (31)

New method option (by default, all three methods are tried in sequence).

$\mathrm{convert}\left(\mathrm{arcsin}{\left(z\right)}^{2},\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{n}{!}}^{{2}}{}{{4}}^{{n}}{}{{z}}^{{2}{}{n}{+}{2}}}{\left({2}{}{n}{+}{2}\right){!}}$ (32)

$\mathrm{convert}\left(\mathrm{arcsin}{\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{hypergeometric}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{n}{!}}^{{2}}{}{{4}}^{{n}}{}{{z}}^{{2}{}{n}{+}{2}}}{\left({2}{}{n}{+}{2}\right){!}}$ (33)

$\mathrm{convert}\left(\mathrm{arcsin}{\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}{+}{1}}{,}{\mathrm{RESol}}{}\left(\left\{\left({-}{{n}}^{{2}}{-}{2}{}{n}{-}{1}\right){}{A}{}\left({n}\right){+}\left({{n}}^{{2}}{+}{5}{}{n}{+}{6}\right){}{A}{}\left({n}{+}{2}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{1}\right\}{,}{\mathrm{INFO}}\right)$ (34)

$\mathrm{convert}\left(\mathrm{arcsin}{\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{4}\right){-}\frac{{4}{}\left({n}{+}{2}\right){}{A}{}\left({n}{+}{2}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{2}}^{{n}}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{3}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right){+}{\sum }_{{\mathrm{_k}}{=}{2}}^{{n}{+}{2}}{}\left({-}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{5}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{5}{-}{\mathrm{_k}}\right)\right)}{{4}{}\left({n}{+}{3}\right)}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{0}{,}{A}{}\left({2}\right){=}{1}{,}{A}{}\left({3}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (35)

$\mathrm{convert}\left(\mathrm{tan}\left(z\right),\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{3}\right){+}\frac{{-}{2}{}{A}{}\left({n}{+}{1}\right){+}{\sum }_{{\mathrm{_k}}{=}{1}}^{{n}}{}\left({-}{2}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({n}{+}{1}{-}{\mathrm{_k}}\right)\right)}{\left({n}{+}{2}\right){}\left({n}{+}{3}\right)}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{1}{,}{A}{}\left({2}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (36)

$\mathrm{convert}\left(\mathrm{tan}\left(z\right),\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{hypergeometric}\right)$

 ${\mathrm{tan}}{}\left({z}\right)$ (37)

$\mathrm{convert}\left(\mathrm{tan}\left(z\right),\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$

 ${\mathrm{tan}}{}\left({z}\right)$ (38)

$\mathrm{convert}\left(\mathrm{tan}\left(z\right),\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{3}\right){+}\frac{{-}{2}{}{A}{}\left({n}{+}{1}\right){+}{\sum }_{{\mathrm{_k}}{=}{1}}^{{n}}{}\left({-}{2}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({n}{+}{1}{-}{\mathrm{_k}}\right)\right)}{\left({n}{+}{2}\right){}\left({n}{+}{3}\right)}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{1}{,}{A}{}\left({2}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (39)

New output option (default: combined).

$\mathrm{convert}\left({\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right)\right)}^{3},\mathrm{FormalPowerSeries}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{n}}{}\left({{9}}^{{n}}{-}{3}\right){}{{z}}^{{2}{}{n}}}{{2}{}\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{3}{}{\left({-1}\right)}^{{n}}{}\left({{9}}^{{n}}{+}{1}\right){}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}\left({2}{}{n}{+}{1}\right){!}}\right)$ (40)

$\mathrm{convert}\left({\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right)\right)}^{3},\mathrm{FormalPowerSeries},\mathrm{output}=\mathrm{combined}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{n}}{}\left({{9}}^{{n}}{-}{3}\right){}{{z}}^{{2}{}{n}}}{{2}{}\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{3}{}{\left({-1}\right)}^{{n}}{}\left({{9}}^{{n}}{+}{1}\right){}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}\left({2}{}{n}{+}{1}\right){!}}\right)$ (41)

$\mathrm{convert}\left({\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right)\right)}^{3},\mathrm{FormalPowerSeries},\mathrm{output}=\mathrm{expanded}\right)$

 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{n}}{}{{9}}^{{n}}{}{{z}}^{{2}{}{n}}}{{2}{}\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{3}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}}}{{2}{}\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{3}{}{\left({-1}\right)}^{{n}}{}{{9}}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}\left({2}{}{n}{+}{1}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{3}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}\left({2}{}{n}{+}{1}\right){!}}\right)$ (42)