Edit detail for TaylorSeries revision 8 of 8

Note 04-Apr-2021: This page is abandoned, it's new home is 
https://fricas.github.io/fricas-notebooks/FriCAS-TaylorSeries.html

Example for multivariate Taylor series expansion
  Univariate TaylorSeries
    FriCAS can deal with power series in a simple manner.
fricas
(1) -> )version

"FriCAS 1.3.12 compiled at Sat  7 Jun 23:54:49 CEST 2025"

fricas
x:=taylor 'x
\begin{equation}
\label{eq1}x\end{equation}
Type: UnivariateTaylorSeries?(Expression(Integer),x,0)
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sinh(x)
\begin{equation}
\label{eq2}\begin{array}{@{}l}
\displaystyle
x +{{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{5
040}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
\end{equation}
Type: UnivariateTaylorSeries?(Expression(Integer),x,0)
    However, sometimes one wants to be more precise with the domain
    that the object lives in. If, for example, we don't want power
    series over the general expression domain as exemplified above,
    we can give the coefficient domain explicitly.

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Z ==> Integer
Type: Void
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Q ==> Fraction Z
Type: Void
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Ux ==> UnivariateTaylorSeries(Q, 'x, 0)
Type: Void
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ux: Ux := 'x
\begin{equation}
\label{eq3}x\end{equation}
Type: UnivariateTaylorSeries?(Fraction(Integer),x,0)
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sinh(ux)
\begin{equation}
\label{eq4}\begin{array}{@{}l}
\displaystyle
x +{{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{5
040}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
\end{equation}
Type: UnivariateTaylorSeries?(Fraction(Integer),x,0)
  Combination of univariate TaylorSeries
    The FriCAS interpreter is smart enough to create an appropriate
    type if two univariate Taylor series interact.
    However, as seen below, the resulting domain is something like
    $$Q[[x]]?[[y]]?,$$
    i.e., univariate power series in y with coefficients that are
    univariate power series in x that have rational coefficients.

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Uy ==> UnivariateTaylorSeries(Q, 'y, 0)
Type: Void
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uy: Uy := 'y
\begin{equation}
\label{eq5}y\end{equation}
Type: UnivariateTaylorSeries?(Fraction(Integer),y,0)
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cosh(uy)
\begin{equation}
\label{eq6}\begin{array}{@{}l}
\displaystyle
1 +{{\frac{1}{2}}\ {{y}^{2}}}+{{\frac{1}{24}}\ {{y}^{4}}}+{{\frac{1}{7
20}}\ {{y}^{6}}}+ 
\
\
\displaystyle
{{\frac{1}{40320}}\ {{y}^{8}}}+{{\frac{1}{3628800}}\ {{y}^{10}}}+{O \left({{y}^{11}}\right)}
\end{array}
\end{equation}
Type: UnivariateTaylorSeries?(Fraction(Integer),y,0)
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sinh(ux)*cosh(uy)
\begin{equation}
\label{eq7}\begin{array}{@{}l}
\displaystyle
x +{{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{5
040}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}+ \
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{2}}\  x}+{{\frac{1}{12}}\ {{x}^{3}}}+{{\frac{1}{24
0}}\ {{x}^{5}}}+ 
\
\
\displaystyle
{{\frac{1}{10080}}\ {{x}^{7}}}+{{\frac{1}{725760}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
}\right)}\ {{y}^{2}}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{24}}\  x}+{{\frac{1}{144}}\ {{x}^{3}}}+{{\frac{1}{2
880}}\ {{x}^{5}}}+ 
\
\
\displaystyle
{{\frac{1}{120960}}\ {{x}^{7}}}+{{\frac{1}{8709120}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
}\right)}\ {{y}^{4}}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{720}}\  x}+{{\frac{1}{4320}}\ {{x}^{3}}}+{{\frac{1}{8
6400}}\ {{x}^{5}}}+ 
\
\
\displaystyle
{{\frac{1}{3628800}}\ {{x}^{7}}}+{{\frac{1}{261273600}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
}\right)}\ {{y}^{6}}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{40320}}\  x}+{{\frac{1}{241920}}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{\frac{1}{4838400}}\ {{x}^{5}}}+{{\frac{1}{203212800}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{14631321600}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
}\right)}\ {{y}^{8}}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{3628800}}\  x}+{{\frac{1}{21772800}}\ {{x}^{3}}}+ \
\
\displaystyle
{{\frac{1}{435456000}}\ {{x}^{5}}}+{{\frac{1}{18289152000}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{1316818944000}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}
\end{array}
}\right)}\ {{y}^{10}}}+ 
\
\
\displaystyle
{O \left({{y}^{11}}\right)}
\end{array}
\end{equation}
Type: UnivariateTaylorSeries?(UnivariateTaylorSeries?(Fraction(Integer),x,0),y,0)
    As a general rule, the FriCAS interpreter tries to find
    a "better coefficient domain" if something does not fit
    into the type of the current object in order to construct
    a more general domain that can hold the result of the operation.
    In the case above that is probably not what we expected or wanted.
  Multivariate TaylorSeries in infinitely many variables
    There is a domain in FriCAS that is similar to the Polynomial(Q) domain.
    TaylorSeries(Q) is the domain of power series over Q in infinitely
    many variables.
    With that domain the input is as simple as for univariate power series.

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T ==> TaylorSeries Q
Type: Void
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tx:T := 'x
\begin{equation}
\label{eq8}x\end{equation}
Type: TaylorSeries(Fraction(Integer))
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ty:T := 'y
\begin{equation}
\label{eq9}y\end{equation}
Type: TaylorSeries(Fraction(Integer))
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sinh(tx)*cosh(ty)
\begin{equation}
\label{eq10}\begin{array}{@{}l}
\displaystyle
x +{\left({{\frac{1}{2}}\  x \ {{y}^{2}}}+{{\frac{1}{6}}\ {{x}^{3}}}\right)}+ 
\
\
\displaystyle
{\left({{\frac{1}{24}}\  x \ {{y}^{4}}}+{{\frac{1}{12}}\ {{x}^{3}}\ {{y}^{2}}}+{{\frac{1}{120}}\ {{x}^{5}}}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{720}}\  x \ {{y}^{6}}}+{{\frac{1}{144}}\ {{x}^{3}}\ {{y}^{4}}}+{{\frac{1}{240}}\ {{x}^{5}}\ {{y}^{2}}}+ 
\
\
\displaystyle
{{\frac{1}{5040}}\ {{x}^{7}}}
\end{array}
}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{40320}}\  x \ {{y}^{8}}}+{{\frac{1}{4320}}\ {{x}^{3}}\ {{y}^{6}}}+ 
\
\
\displaystyle
{{\frac{1}{2880}}\ {{x}^{5}}\ {{y}^{4}}}+{{\frac{1}{10080}}\ {{x}^{7}}\ {{y}^{2}}}+ 
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}
\end{array}
}\right)}+ 
\
\
\displaystyle
{O \left({11}\right)}
\end{array}
\end{equation}
Type: TaylorSeries(Fraction(Integer))
  Multivariate TaylorSeries in two variables
    FriCAS allows to be more precise with multivariate power series.
    It is possible to create multivariate power series in a given
    number of variables. Such a construction is, however, a bit more
    involved.
    The domain (named M below') is modelled as a univariate power series
    over bivariate polynomials where the n-th coefficient of the series is
    the polynomial consisting of all (bivariate) terms of degree n.
    Thus we first have to create a bivariate polynomial domain.
    From this construction, it should be clear how to create multivariate
    series in three or more variables.

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vl: List Symbol := ['x, 'y]
\begin{equation*}
\label{eq11}\left[ x , \: y \right]?\end{equation*}
Type: List(Symbol)
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V ==> OrderedVariableList vl
Type: Void
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P ==> SparseMultivariatePolynomial(Q, V)
Type: Void
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M ==> SparseMultivariateTaylorSeries(Q, V, P)
Type: Void
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X: M := monomial(1$M, 'x, 1)
\begin{equation}
\label{eq12}x\end{equation}
Type: SparseMultivariateTaylorSeries(Fraction(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList([x,y])))
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Y: M := monomial(1$M, 'y, 1)
\begin{equation}
\label{eq13}y\end{equation}
Type: SparseMultivariateTaylorSeries(Fraction(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList([x,y])))
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sinh(X)*cosh(Y)
\begin{equation}
\label{eq14}\begin{array}{@{}l}
\displaystyle
x +{\left({{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{2}}\ {{y}^{2}}\  x}\right)}+ 
\
\
\displaystyle
{\left({{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{12}}\ {{y}^{2}}\ {{x}^{3}}}+{{\frac{1}{24}}\ {{y}^{4}}\  x}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{5040}}\ {{x}^{7}}}+{{\frac{1}{240}}\ {{y}^{2}}\ {{x}^{5}}}+{{\frac{1}{144}}\ {{y}^{4}}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{\frac{1}{720}}\ {{y}^{6}}\  x}
\end{array}
}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{{\frac{1}{10080}}\ {{y}^{2}}\ {{x}^{7}}}+ 
\
\
\displaystyle
{{\frac{1}{2880}}\ {{y}^{4}}\ {{x}^{5}}}+{{\frac{1}{4320}}\ {{y}^{6}}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{\frac{1}{40320}}\ {{y}^{8}}\  x}
\end{array}
}\right)}+ 
\
\
\displaystyle
{O \left({11}\right)}
\end{array}
\end{equation}
Type: SparseMultivariateTaylorSeries(Fraction(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList([x,y])))

Multivariate Taylor series with unknown coefficients
    We want to generate taylor series with unknown coefficients.
fricas
)clear completely

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.
vl: List Symbol := ['x, 'y];
Type: List(Symbol)
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V ==> OrderedVariableList vl
Type: Void
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Q ==> Expression Integer
Type: Void
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P ==> SparseMultivariatePolynomial(Q, V)
Type: Void
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M ==> SparseMultivariateTaylorSeries(Q, V, P)
Type: Void
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X: M := monomial(1$M, 'x, 1)
\begin{equation}
\label{eq15}x\end{equation}
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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Y: M := monomial(1$M, 'y, 1)
\begin{equation}
\label{eq16}y\end{equation}
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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sx:M := recip(1-X)
\begin{equation}
\label{eq17}1 + x +{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+{{x}^{5}}+{{x}^{6}}+{{x}^{7}}+{{x}^{8}}+{{x}^{9}}+{{x}^{10}}+{O \left({11}\right)}\end{equation}
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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sy:M := recip(1-Y)
\begin{equation}
\label{eq18}1 + y +{{y}^{2}}+{{y}^{3}}+{{y}^{4}}+{{y}^{5}}+{{y}^{6}}+{{y}^{7}}+{{y}^{8}}+{{y}^{9}}+{{y}^{10}}+{O \left({11}\right)}\end{equation}
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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s1 := sx*sy
\begin{equation}
\label{eq19}\begin{array}{@{}l}
\displaystyle
1 +{\left(x + y \right)}+{\left({{x}^{2}}+{y \  x}+{{y}^{2}}\right)}+{\left({{x}^{3}}+{y \ {{x}^{2}}}+{{{y}^{2}}\  x}+{{y}^{3}}\right)}+ 
\
\
\displaystyle
{\left({{x}^{4}}+{y \ {{x}^{3}}}+{{{y}^{2}}\ {{x}^{2}}}+{{{y}^{3}}\  x}+{{y}^{4}}\right)}+ 
\
\
\displaystyle
{\left({{x}^{5}}+{y \ {{x}^{4}}}+{{{y}^{2}}\ {{x}^{3}}}+{{{y}^{3}}\ {{x}^{2}}}+{{{y}^{4}}\  x}+{{y}^{5}}\right)}+ 
\
\
\displaystyle
{\left({{x}^{6}}+{y \ {{x}^{5}}}+{{{y}^{2}}\ {{x}^{4}}}+{{{y}^{3}}\ {{x}^{3}}}+{{{y}^{4}}\ {{x}^{2}}}+{{{y}^{5}}\  x}+{{y}^{6}}\right)}+ 
\
\
\displaystyle
{\left({{x}^{7}}+{y \ {{x}^{6}}}+{{{y}^{2}}\ {{x}^{5}}}+{{{y}^{3}}\ {{x}^{4}}}+{{{y}^{4}}\ {{x}^{3}}}+{{{y}^{5}}\ {{x}^{2}}}+{{{y}^{6}}\  x}+{{y}^{7}}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{x}^{8}}+{y \ {{x}^{7}}}+{{{y}^{2}}\ {{x}^{6}}}+{{{y}^{3}}\ {{x}^{5}}}+{{{y}^{4}}\ {{x}^{4}}}+{{{y}^{5}}\ {{x}^{3}}}+{{{y}^{6}}\ {{x}^{2}}}+ 
\
\
\displaystyle
{{{y}^{7}}\  x}+{{y}^{8}}
\end{array}
}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{x}^{9}}+{y \ {{x}^{8}}}+{{{y}^{2}}\ {{x}^{7}}}+{{{y}^{3}}\ {{x}^{6}}}+{{{y}^{4}}\ {{x}^{5}}}+{{{y}^{5}}\ {{x}^{4}}}+{{{y}^{6}}\ {{x}^{3}}}+ 
\
\
\displaystyle
{{{y}^{7}}\ {{x}^{2}}}+{{{y}^{8}}\  x}+{{y}^{9}}
\end{array}
}\right)}+ 
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{x}^{10}}+{y \ {{x}^{9}}}+{{{y}^{2}}\ {{x}^{8}}}+{{{y}^{3}}\ {{x}^{7}}}+{{{y}^{4}}\ {{x}^{6}}}+{{{y}^{5}}\ {{x}^{5}}}+{{{y}^{6}}\ {{x}^{4}}}+ 
\
\
\displaystyle
{{{y}^{7}}\ {{x}^{3}}}+{{{y}^{8}}\ {{x}^{2}}}+{{{y}^{9}}\  x}+{{y}^{10}}
\end{array}
}\right)}+ 
\
\
\displaystyle
{O \left({11}\right)}
\end{array}
\end{equation}
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
    We can create power series with
    unknown coefficients.

fricas
a: Symbol := 'a;
Type: Symbol
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fp(p:P):P == (pp:P := 0; e:=enumerate()$V; for m in monomials p repeat (l:=degree(m,e); pp := pp + elt(a,l)*m);pp)

   Function declaration fp : SparseMultivariatePolynomial(Expression(
      Integer),OrderedVariableList([x,y])) -> 
      SparseMultivariatePolynomial(Expression(Integer),
      OrderedVariableList([x,y])) has been added to workspace.
Type: Void
fricas
st1: Stream P := coefficients s1;
Type: Stream(SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
fricas
ast1: Stream P := map(fp, st1);
fricas
Compiling function fp with type SparseMultivariatePolynomial(
      Expression(Integer),OrderedVariableList([x,y])) -> 
      SparseMultivariatePolynomial(Expression(Integer),
      OrderedVariableList([x,y]))
Type: Stream(SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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a1: M := series ast1;
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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t:=(X+Y-1)*a1;
Type: SparseMultivariateTaylorSeries(Expression(Integer),OrderedVariableList([x,y]),SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y])))
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coefficient(t,3)
\begin{equation}
\label{eq20}\begin{array}{@{}l}
\displaystyle
{{\left(-{a_{3, \: 0}}+{a_{2, \: 0}}\right)}\ {{x}^{3}}}+{{\left(-{a_{2, \: 1}}+{a_{2, \: 0}}+{a_{1, \: 1}}\right)}\  y \ {{x}^{2}}}+ 
\
\
\displaystyle
{{\left(-{a_{1, \: 2}}+{a_{1, \: 1}}+{a_{0, \: 2}}\right)}\ {{y}^{2}}\  x}+{{\left(-{a_{0, \: 3}}+{a_{0, \: 2}}\right)}\ {{y}^{3}}}
\end{array}
\end{equation}
Type: SparseMultivariatePolynomial?(Expression(Integer),OrderedVariableList([x,y]))
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c := concat [coefficients coefficient(t, n) for n in 0..4]
\begin{equation*}
\label{eq21}\begin{array}{@{}l}
\displaystyle
\left[ -{a_{0, \: 0}}, \:{-{a_{1, \: 0}}+{a_{0, \: 0}}}, \:{-{a_{0, \: 1}}+{a_{0, \: 0}}}, \:{-{a_{2, \: 0}}+{a_{1, \: 0}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{1, \: 1}}+{a_{1, \: 0}}+{a_{0, \: 1}}}, \:{-{a_{0, \: 2}}+{a_{0, \: 1}}}, \:{-{a_{3, \: 0}}+{a_{2, \: 0}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{2, \: 1}}+{a_{2, \: 0}}+{a_{1, \: 1}}}, \:{-{a_{1, \: 2}}+{a_{1, \: 1}}+{a_{0, \: 2}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{0, \: 3}}+{a_{0, \: 2}}}, \:{-{a_{4, \: 0}}+{a_{3, \: 0}}}, \:{-{a_{3, \: 1}}+{a_{3, \: 0}}+{a_{2, \: 1}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{2, \: 2}}+{a_{2, \: 1}}+{a_{1, \: 2}}}, \:{-{a_{1, \: 3}}+{a_{1, \: 2}}+{a_{0, \: 3}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{0, \: 4}}+{a_{0, \: 3}}}\right] 
\end{array}
\end{equation*}
Type: List(Expression(Integer))
fricas
variables first c
\begin{equation*}
\label{eq22}\left[{a_{0, \: 0}}\right]?\end{equation*}
Type: List(Symbol)
fricas
vars: List Symbol := concat [variables z for z in c]
\begin{equation*}
\label{eq23}\begin{array}{@{}l}
\displaystyle
\left[{a_{0, \: 0}}, \:{a_{0, \: 0}}, \:{a_{1, \: 0}}, \:{a_{0, \: 0}}, \:{a_{0, \: 1}}, \:{a_{1, \: 0}}, \:{a_{2, \: 0}}, \:{a_{0, \: 1}}, \:{a_{1, \: 0}}, \:{a_{1, \: 1}}, \: \right.
\
\
\displaystyle
\left.{a_{0, \: 1}}, \:{a_{0, \: 2}}, \:{a_{2, \: 0}}, \:{a_{3, \: 0}}, \:{a_{1, \: 1}}, \:{a_{2, \: 0}}, \:{a_{2, \: 1}}, \:{a_{0, \: 2}}, \:{a_{1, \: 1}}, \:{a_{1, \: 2}}, \: \right.
\
\
\displaystyle
\left.{a_{0, \: 2}}, \:{a_{0, \: 3}}, \:{a_{3, \: 0}}, \:{a_{4, \: 0}}, \:{a_{2, \: 1}}, \:{a_{3, \: 0}}, \:{a_{3, \: 1}}, \:{a_{1, \: 2}}, \:{a_{2, \: 1}}, \:{a_{2, \: 2}}, \: \right.
\
\
\displaystyle
\left.{a_{0, \: 3}}, \:{a_{1, \: 2}}, \:{a_{1, \: 3}}, \:{a_{0, \: 3}}, \:{a_{0, \: 4}}\right] 
\end{array}
\end{equation*}
Type: List(Symbol)
fricas
v: List Symbol := [u for u in members set vars]
\begin{equation*}
\label{eq24}\begin{array}{@{}l}
\displaystyle
\left[{a_{0, \: 0}}, \:{a_{0, \: 1}}, \:{a_{0, \: 2}}, \:{a_{0, \: 3}}, \:{a_{0, \: 4}}, \:{a_{1, \: 0}}, \:{a_{1, \: 1}}, \:{a_{1, \: 2}}, \:{a_{1, \: 3}}, \:{a_{2, \: 0}}, \: \right.
\
\
\displaystyle
\left.{a_{2, \: 1}}, \:{a_{2, \: 2}}, \:{a_{3, \: 0}}, \:{a_{3, \: 1}}, \:{a_{4, \: 0}}\right] 
\end{array}
\end{equation*}
Type: List(Symbol)
fricas
es:=cons(a[0,0]-1, rest c)
\begin{equation*}
\label{eq25}\begin{array}{@{}l}
\displaystyle
\left[{{a_{0, \: 0}}- 1}, \:{-{a_{1, \: 0}}+{a_{0, \: 0}}}, \:{-{a_{0, \: 1}}+{a_{0, \: 0}}}, \:{-{a_{2, \: 0}}+{a_{1, \: 0}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{1, \: 1}}+{a_{1, \: 0}}+{a_{0, \: 1}}}, \:{-{a_{0, \: 2}}+{a_{0, \: 1}}}, \:{-{a_{3, \: 0}}+{a_{2, \: 0}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{2, \: 1}}+{a_{2, \: 0}}+{a_{1, \: 1}}}, \:{-{a_{1, \: 2}}+{a_{1, \: 1}}+{a_{0, \: 2}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{0, \: 3}}+{a_{0, \: 2}}}, \:{-{a_{4, \: 0}}+{a_{3, \: 0}}}, \:{-{a_{3, \: 1}}+{a_{3, \: 0}}+{a_{2, \: 1}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{2, \: 2}}+{a_{2, \: 1}}+{a_{1, \: 2}}}, \:{-{a_{1, \: 3}}+{a_{1, \: 2}}+{a_{0, \: 3}}}, \: \right.
\
\
\displaystyle
\left.{-{a_{0, \: 4}}+{a_{0, \: 3}}}\right] 
\end{array}
\end{equation*}
Type: List(Expression(Integer))
    Unfortunately, removing the semicolon from the end of
    the following command makes trouble for LaTeX.

fricas
result:=solve([e=0 for e in es], v);
Type: List(List(Equation(Expression(Integer))))
    Bill Page: It seems to be a problem with encoding a list of lists.

fricas
for i in result repeat outputAsTex(i)
\begin{equation}
\label{eq26}\begin{array}{@{}l}
\displaystyle
{{a_{0, \: 0}}= 1}\ {{a_{0, \: 1}}= 1}\ {{a_{0, \: 2}}= 1}\ {{a_{0, \: 3}}= 1}\ {{a_{0, \: 4}}= 1}\ {{a_{1, \: 0}}= 1}\  \cdot \
\
\displaystyle
{{a_{1, \: 1}}= 2}\ {{a_{1, \: 2}}= 3}\ {{a_{1, \: 3}}= 4}\ {{a_{2, \: 0}}= 1}\ {{a_{2, \: 1}}= 3}\ {{a_{2, \: 2}}= 6}\  \cdot \
\
\displaystyle
{{a_{3, \: 0}}= 1}\ {{a_{3, \: 1}}= 4}\ {{a_{4, \: 0}}= 1}
\end{array}
\end{equation}
Type: Void

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