Convert to Formal Power Series - Maple Help

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convert/FormalPowerSeries

convert to formal Laurent-Puiseux series

 Calling Sequence convert(f, FormalPowerSeries, eq, n, opts) convert(f, FormalPowerSeries, eq, a(n), opts) convert(f, FPS, eq, n, opts) convert(f, FPS, eq, a(n), opts)

Parameters

 f - algebraic expression eq - equation (e.g. z=z0) or name (e.g. z); optional if f contains only one variable n - (optional) name of the summation variable in the output a(n) - (optional) name for the nth series coefficient in the output, which will be a recursive formula for these coefficients instead of a formal power series opts - sequence of optional equations of the form keyword=value; possible keywords are method, differentialorder, recurrence, or output

Returns

 • When a hypergeometric type representation is returned, it can either be a single inert Sum, or a linear combination of such Sums, possibly plus some initial terms that are linear combinations of rational powers of $z-\mathrm{z0}$ or $\frac{1}{z}$.
 • When such a representation cannot be obtained, or when a recursive formula is requested, the output is a sequence of two expressions. The first return value is similar to the representation described above, except that the coefficients of one of the Sums are indeterminate, such as, e.g., ${\sum }_{n=0}^{\mathrm{\infty }}a\left(n\right){z}^{n}$. The second return value is an REsol data structure giving a recurrence relation for $a\left(n\right)$, including initial conditions.

Options

 • differentialorder: positive integer $N$ or list of two positive integers $\left[{N}_{1},{N}_{2}\right]$; upper bound(s) for the order of the differential equation(s) (DE) internally computed for f. Default: ${N}_{1}=10$ for holonomic DEs and ${N}_{2}=4$ for quadratic DEs. If differentialorder=$N$ is given, then the upper bound for holonomic DEs is ${N}_{1}=\mathrm{max}\left(10,N\right)$, and the upper bound ${N}_{2}$ for quadratic DEs is $4$ if $N\le 4$, $9$ if $N\ge 9$, and $N$ otherwise.
 • method: one of hypergeometric, holonomic, quadratic, or default; specifies the underlying computational method that will be used. method=hypergeometric looks for a closed-form formula of the representation whereas holonomic and quadratic look for a recursive formula from a holonomic or a quadratic differential equation satisfied by f. The default method applies the three methods one after the other and stops once a result is found.
 • recurrence: either true or false (default). If recurrence=true then a recursive formula with the coefficient a(n) is sought. This reduces the methods applied to only the two options holonomic and/or quadratic. If a(n) is specified as the 4th argument, then the default is recurrence=true. If both options method and recurrence are given, the former takes precedence.
 • output: expanded or combined (default); controls the presentation of hypergeometric type power series (method=hypergeometric). output=combined gathers series of same power in one summation, whereas output=expanded puts linearly (rationally) independent coefficients in different summations.

Description

 • This command computes the power series representation of linear combinations of meromorphic functions and some special functions.
 • convert(f, FormalPowerSeries, z=z0), or, equivalently, convert(f, FPS, z=z0), tries to find a formal power series expansion for f with respect to the variable z at the point of expansion z0.
 • If eq is just a name z, then the expansion point $z=0$ is assumed.
 • The output series is a hypergeometric type power series, or a recursive formula of the representation sought when a hypergeometric type representation cannot be computed. The user may also specify the desired representation using the method option.
 • See the references for a precise definition of what is called hypergeometric type power series. It generalizes the family of Laurent-Puiseux series having hypergeometric term coefficients.
 • The command first looks for a holonomic differential equation satisfied by f (see DEtools[FindODE]); hence Maple must know the derivatives of f.
 • If the summation variable n is not specified, then one of of the variable names n, n0, n1, etc. is chosen.
 • For a complete list of known functions, see inifcns.

Examples

 > $\mathrm{convert}\left(\mathrm{cos}\left(z\right),\mathrm{FormalPowerSeries},z=\frac{\mathrm{Pi}}{2}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{n}}{}{\left({z}{-}\frac{{\mathrm{\pi }}}{{2}}\right)}^{{2}{}{n}{+}{1}}}{\left({2}{}{n}{+}{1}\right){!}}\right)$ (1)
 > $\mathrm{convert}\left({ⅇ}^{x}+\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],{x}^{2}\right),\mathrm{FormalPowerSeries},x,k\right)$
 $\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{x}}^{{k}}}{{k}{!}}\right){+}\left({\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{pochhammer}}{}\left({b}{,}{k}\right){}{\mathrm{pochhammer}}{}\left({a}{,}{k}\right){}{{x}}^{{2}{}{k}}}{{k}{!}{}{\mathrm{pochhammer}}{}\left({c}{,}{k}\right)}\right)$ (2)
 > $\mathrm{convert}\left(\mathrm{ln}\left(1+\sqrt{z}+z+{z}^{\frac{3}{2}}\right),\mathrm{FormalPowerSeries}\right)$
 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{\frac{{n}}{{2}}{+}\frac{{1}}{{2}}}}{{n}{+}{1}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{{n}{+}{1}}}{{n}{+}{1}}\right)$ (3)
 > $\mathrm{convert}\left(z\mathrm{cos}\left({z}^{\frac{3}{2}}\right)+{\mathrm{arcsin}\left({z}^{\frac{1}{3}}\right)}^{2},\mathrm{FormalPowerSeries}\right)$
 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{4}}^{{n}}{}{{n}{!}}^{{2}}{}{{z}}^{\frac{{2}{}{n}}{{3}}{+}\frac{{2}}{{3}}}}{\left({2}{}{n}{+}{2}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{n}}{}{{z}}^{{3}{}{n}{+}{1}}}{\left({2}{}{n}\right){!}}\right)$ (4)
 > $\mathrm{convert}\left(\frac{1}{\left({z}^{2}-p\right)\left({z}^{3}-q\right)},\mathrm{FormalPowerSeries},z\right)$
 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({-}{{p}}^{{2}{-}\frac{{n}}{{2}}}{+}{2}{}{{q}}^{{-}\frac{{1}}{{3}}{-}\frac{{n}}{{3}}}{}{{p}}^{{5}}{{2}}}{+}{\left({-1}\right)}^{{n}}{}{{p}}^{{2}{-}\frac{{n}}{{2}}}{-}{2}{}{q}{}{{p}}^{{-}\frac{{n}}{{2}}{+}\frac{{1}}{{2}}}{-}{\left({-1}\right)}^{{n}}{}{{p}}^{{-}{1}{-}\frac{{n}}{{2}}}{}{{q}}^{{2}}{+}{2}{}{p}{}{{q}}^{\frac{{2}}{{3}}{-}\frac{{n}}{{3}}}{-}{{p}}^{{-}{1}{-}\frac{{n}}{{2}}}{}{{q}}^{{2}}\right){}{{z}}^{{n}}}{{2}{}\left({{p}}^{{3}}{-}{{q}}^{{2}}\right){}\left({{p}}^{{3}}{{2}}}{+}{q}\right)}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{p}{}{{q}}^{{-}{1}{-}{n}}{}{{z}}^{{3}{}{n}}}{{{p}}^{{2}}{+}{{q}}^{{2}}{{3}}}{}{p}{+}{{q}}^{{4}}{{3}}}}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{{q}}^{{-}{n}{-}\frac{{2}}{{3}}}{}{{z}}^{{3}{}{n}{+}{1}}}{{{p}}^{{2}}{+}{{q}}^{{2}}{{3}}}{}{p}{+}{{q}}^{{4}}{{3}}}}\right)$ (5)

Illustration for the output option

 > $\mathrm{convert}\left({\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right)\right)}^{5},\mathrm{FormalPowerSeries}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{\left({{25}}^{{n}}{+}{5}{}{{9}}^{{n}}{-}{10}\right){}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}}}{{4}{}\left({2}{}{n}\right){!}}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{5}{}\left({{25}}^{{n}}{-}{3}{}{{9}}^{{n}}{-}{2}\right){}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{4}{}\left({2}{}{n}{+}{1}\right){!}}\right)$ (6)
 > $\mathrm{convert}\left({\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right)\right)}^{5},\mathrm{FormalPowerSeries},\mathrm{output}=\mathrm{expanded}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\left({-1}\right)}^{{n}}{}{{25}}^{{n}}{}{{z}}^{{2}{}{n}}}{{4}{}\left({2}{}{n}\right){!}}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{5}{}{\left({-1}\right)}^{{n}}{}{{9}}^{{n}}{}{{z}}^{{2}{}{n}}}{{4}{}\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{5}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}}}{{2}{}\left({2}{}{n}\right){!}}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{5}{}{\left({-1}\right)}^{{n}}{}{{25}}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{4}{}\left({2}{}{n}{+}{1}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{15}{}{\left({-1}\right)}^{{n}}{}{{9}}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{4}{}\left({2}{}{n}{+}{1}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{5}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{{2}{}\left({2}{}{n}{+}{1}\right){!}}\right)$ (7)

Illustration for the method option. All the outputs below can be obtained by the default option. However, one can directly specify the method that yields the result in order to save time. ${ⅇ}^{{z}^{3}+z}$ and ${\mathrm{arcsin}\left(z\right)}^{3}$  are holonomic but not of hypergeometric type.

 > $\mathrm{convert}\left({ⅇ}^{z+{z}^{3}},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left(\left\{\left({n}{+}{3}\right){}{A}{}\left({n}{+}{3}\right){-}{3}{}{A}{}\left({n}\right){-}{A}{}\left({n}{+}{2}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{1}{,}{A}{}\left({1}\right){=}{1}{,}{A}{}\left({2}\right){=}\frac{{1}}{{2}}\right\}{,}{\mathrm{INFO}}\right)$ (8)
 > $\mathrm{convert}\left({\mathrm{arcsin}\left(z\right)}^{3},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}{+}{1}}{,}{\mathrm{RESol}}{}\left(\left\{\left({{n}}^{{4}}{+}{4}{}{{n}}^{{3}}{+}{6}{}{{n}}^{{2}}{+}{4}{}{n}{+}{1}\right){}{A}{}\left({n}\right){+}\left({-}{2}{}{{n}}^{{4}}{-}{18}{}{{n}}^{{3}}{-}{62}{}{{n}}^{{2}}{-}{98}{}{n}{-}{60}\right){}{A}{}\left({n}{+}{2}\right){+}\left({{n}}^{{4}}{+}{14}{}{{n}}^{{3}}{+}{71}{}{{n}}^{{2}}{+}{154}{}{n}{+}{120}\right){}{A}{}\left({n}{+}{4}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{0}{,}{A}{}\left({2}\right){=}{1}{,}{A}{}\left({3}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (9)

$\frac{z}{{ⅇ}^{z}-1}$ and $1+{\mathrm{tan}\left(z\right)}^{2}$ are not holonomic. Bernoulli numbers:

 > $\mathrm{convert}\left(\frac{z}{{ⅇ}^{z}-1},\mathrm{FormalPowerSeries},B\left(n\right),\mathrm{method}=\mathrm{quadratic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{B}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({B}{}\left({n}{+}{3}\right){+}\frac{\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{2}}{}{B}{}\left({\mathrm{_k}}\right){}{B}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right){+}{B}{}\left({n}{+}{2}\right)}{{n}{+}{4}}{,}\left\{{B}{}\left({n}\right)\right\}{,}\left\{{B}{}\left({0}\right){=}{1}{,}{B}{}\left({1}\right){=}{-}\frac{{1}}{{2}}{,}{B}{}\left({2}\right){=}\frac{{1}}{{12}}\right\}{,}{\mathrm{INFO}}\right)$ (10)
 > $\mathrm{convert}\left(1+{\mathrm{tan}\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{2}\right){+}\frac{{-}{8}{}{A}{}\left({n}\right){+}{\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{-}{1}}{}\left({-}{6}{}{A}{}\left({\mathrm{_k}}\right){}{A}{}\left({n}{-}{\mathrm{_k}}\right)\right)}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{1}{,}{A}{}\left({1}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (11)

Different results from the three methods

 > $\mathrm{convert}\left({\mathrm{arcsin}\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{hypergeometric}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{4}}^{{n}}{}{{n}{!}}^{{2}}{}{{z}}^{{2}{}{n}{+}{2}}}{\left({2}{}{n}{+}{2}\right){!}}$ (12)
 > $\mathrm{convert}\left({\mathrm{arcsin}\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}{+}{1}}{,}{\mathrm{RESol}}{}\left(\left\{\left({-}{{n}}^{{2}}{-}{2}{}{n}{-}{1}\right){}{A}{}\left({n}\right){+}\left({{n}}^{{2}}{+}{5}{}{n}{+}{6}\right){}{A}{}\left({n}{+}{2}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{1}\right\}{,}{\mathrm{INFO}}\right)$ (13)
 > $\mathrm{convert}\left({\mathrm{arcsin}\left(z\right)}^{2},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{4}\right){-}\frac{{4}{}\left({n}{+}{2}\right){}{A}{}\left({n}{+}{2}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{2}}^{{n}}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{3}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right){+}{\sum }_{{\mathrm{_k}}{=}{2}}^{{n}{+}{2}}{}\left({-}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{5}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{5}{-}{\mathrm{_k}}\right)\right)}{{4}{}\left({n}{+}{3}\right)}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{0}{,}{A}{}\left({1}\right){=}{0}{,}{A}{}\left({2}\right){=}{1}{,}{A}{}\left({3}\right){=}{0}\right\}{,}{\mathrm{INFO}}\right)$ (14)
 > $\mathrm{convert}\left({ⅇ}^{\mathrm{arcsin}\left(z\right)},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{hypergeometric}\right)$
 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{pochhammer}}{}\left({-}\frac{{I}}{{2}}{,}{n}\right){}{\mathrm{pochhammer}}{}\left(\frac{{I}}{{2}}{,}{n}\right){}{{4}}^{{n}}{}{{z}}^{{2}{}{n}}}{\left({2}{}{n}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{pochhammer}}{}\left(\frac{{1}}{{2}}{-}\frac{{I}}{{2}}{,}{n}\right){}{\mathrm{pochhammer}}{}\left(\frac{{1}}{{2}}{+}\frac{{I}}{{2}}{,}{n}\right){}{{4}}^{{n}}{}{{z}}^{{2}{}{n}{+}{1}}}{\left({2}{}{n}{+}{1}\right){!}}\right)$ (15)
 > $\mathrm{convert}\left({ⅇ}^{\mathrm{arcsin}\left(z\right)},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{holonomic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left(\left\{\left({-}{{n}}^{{2}}{-}{1}\right){}{A}{}\left({n}\right){+}\left({{n}}^{{2}}{+}{3}{}{n}{+}{2}\right){}{A}{}\left({n}{+}{2}\right){=}{0}\right\}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{1}{,}{A}{}\left({1}\right){=}{1}\right\}{,}{\mathrm{INFO}}\right)$ (16)
 > $\mathrm{convert}\left({ⅇ}^{\mathrm{arcsin}\left(z\right)},\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{A}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({A}{}\left({n}{+}{3}\right){+}\frac{{2}{}\left({n}{+}{1}\right){}{A}{}\left({n}{+}{1}\right){+}{2}{}{A}{}\left({n}{+}{2}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{-}{1}}{}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{1}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{1}{-}{\mathrm{_k}}\right)\right){+}{\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{1}}{}\left({-}\left({\mathrm{_k}}{+}{1}\right){}{A}{}\left({\mathrm{_k}}{+}{1}\right){}\left({n}{+}{3}{-}{\mathrm{_k}}\right){}{A}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{1}}{}{A}{}\left({\mathrm{_k}}\right){}{A}{}\left({n}{+}{2}{-}{\mathrm{_k}}\right)\right)}{{-}{2}{}{n}{-}{6}}{,}\left\{{A}{}\left({n}\right)\right\}{,}\left\{{A}{}\left({0}\right){=}{1}{,}{A}{}\left({1}\right){=}{1}{,}{A}{}\left({2}\right){=}\frac{{1}}{{2}}\right\}{,}{\mathrm{INFO}}\right)$ (17)

The quadratic method detects non-trivial identities

 > $f≔2\mathrm{arctanh}\left(\frac{\mathrm{sin}\left(2z\right)}{1+\mathrm{cos}\left(2z\right)}\right):$$g≔\mathrm{ln}\left(\frac{1+\mathrm{tan}\left(z\right)}{1-\mathrm{tan}\left(z\right)}\right):$
 > $\mathrm{convert}\left(f-g,\mathrm{FormalPowerSeries},\mathrm{method}=\mathrm{quadratic}\right)$
 ${0}$ (18)

Illustration for the recurrence option. One of $\mathrm{holonomic}$ and $\mathrm{quadratic}$ is used for the method.

 > $\mathrm{convert}\left(\mathrm{sin}\left(z\right)+\mathrm{cos}\left(z\right),\mathrm{FormalPowerSeries},a\left(n\right),\mathrm{recurrence}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{a}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left(\left\{{a}{}\left({n}\right){+}\left({{n}}^{{2}}{+}{3}{}{n}{+}{2}\right){}{a}{}\left({n}{+}{2}\right){=}{0}\right\}{,}\left\{{a}{}\left({n}\right)\right\}{,}\left\{{a}{}\left({0}\right){=}{1}{,}{a}{}\left({1}\right){=}{1}\right\}{,}{\mathrm{INFO}}\right)$ (19)
 > $\mathrm{convert}\left({\mathrm{ln}\left(1+z\right)}^{-1},\mathrm{FormalPowerSeries},a\left(n\right),\mathrm{recurrence}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{a}{}\left({n}\right){}{{z}}^{{n}{-}{1}}{,}{\mathrm{RESol}}{}\left({a}{}\left({n}{+}{3}\right){+}\frac{\left({n}{+}{1}\right){}{a}{}\left({n}{+}{2}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}{+}{2}}{}{a}{}\left({\mathrm{_k}}\right){}{a}{}\left({n}{+}{3}{-}{\mathrm{_k}}\right)\right)}{{n}{+}{4}}{,}\left\{{a}{}\left({n}\right)\right\}{,}\left\{{a}{}\left({0}\right){=}{1}{,}{a}{}\left({1}\right){=}\frac{{1}}{{2}}{,}{a}{}\left({2}\right){=}{-}\frac{{1}}{{12}}\right\}{,}{\mathrm{INFO}}\right)$ (20)
 > $\mathrm{convert}\left(\mathrm{ln}\left(1+\mathrm{sin}\left(z\right)\right),\mathrm{FormalPowerSeries},a\left(n\right),\mathrm{recurrence}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{a}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left({a}{}\left({n}{+}{4}\right){+}\frac{{-}\left({n}{+}{2}\right){}{a}{}\left({n}{+}{2}\right){+}\left({n}{+}{2}\right){}\left({n}{+}{3}\right){}{a}{}\left({n}{+}{3}\right){+}\left({\sum }_{{\mathrm{_k}}{=}{1}}^{{n}}{}\left({\mathrm{_k}}{+}{1}\right){}\left({\mathrm{_k}}{+}{2}\right){}{a}{}\left({\mathrm{_k}}{+}{2}\right){}\left({n}{+}{2}{-}{\mathrm{_k}}\right){}{a}{}\left({n}{+}{2}{-}{\mathrm{_k}}\right)\right)}{\left({n}{+}{2}\right){}\left({n}{+}{3}\right){}\left({n}{+}{4}\right)}{,}\left\{{a}{}\left({n}\right)\right\}{,}\left\{{a}{}\left({0}\right){=}{0}{,}{a}{}\left({1}\right){=}{1}{,}{a}{}\left({2}\right){=}{-}\frac{{1}}{{2}}{,}{a}{}\left({3}\right){=}\frac{{1}}{{6}}\right\}{,}{\mathrm{INFO}}\right)$ (21)
 > $\mathrm{convert}\left(\mathrm{arctan}\left(z\right){ⅇ}^{z},\mathrm{FormalPowerSeries},a\left(n\right),\mathrm{recurrence}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{a}{}\left({n}\right){}{{z}}^{{n}}{,}{\mathrm{RESol}}{}\left(\left\{{a}{}\left({n}\right){+}\left({-}{2}{}{n}{-}{4}\right){}{a}{}\left({n}{+}{1}\right){+}\left({{n}}^{{2}}{+}{5}{}{n}{+}{7}\right){}{a}{}\left({n}{+}{2}\right){+}\left({-}{2}{}{n}{-}{6}\right){}{a}{}\left({n}{+}{3}\right){+}\left({{n}}^{{2}}{+}{7}{}{n}{+}{12}\right){}{a}{}\left({n}{+}{4}\right){=}{0}\right\}{,}\left\{{a}{}\left({n}\right)\right\}{,}\left\{{a}{}\left({0}\right){=}{0}{,}{a}{}\left({1}\right){=}{1}{,}{a}{}\left({2}\right){=}{1}{,}{a}{}\left({3}\right){=}\frac{{1}}{{6}}\right\}{,}{\mathrm{INFO}}\right)$ (22)

More examples.

 > $\mathrm{convert}\left({ⅇ}^{z}\mathrm{ln}\left(z\right),\mathrm{FormalPowerSeries}\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{ln}}{}\left({z}\right){}{{z}}^{{n}}}{{n}{!}}$ (23)
 > $\mathrm{convert}\left({ⅇ}^{z}\mathrm{sin}\left(z\right),\mathrm{FormalPowerSeries}\right)$
 $\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{4}}^{{n}}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{4}{}{n}{+}{1}}}{\left({4}{}{n}{+}{1}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{4}}^{{n}}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{4}{}{n}{+}{2}}}{\left({4}{}{n}{+}{2}\right){!}}\right){+}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{4}}^{{n}}{}{\left({-1}\right)}^{{n}}{}{{z}}^{{4}{}{n}{+}{3}}}{\left({4}{}{n}{+}{3}\right){!}}\right)$ (24)
 > $\mathrm{convert}\left(\frac{1}{p+qz+r{z}^{3}},\mathrm{FormalPowerSeries},z\right)$