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 dAlembertian_series_sol
 formal power series solutions with d'Alembertian coefficients for a linear ODE

 Calling Sequence dAlembertian_series_sol(ode,var,opts) dAlembertian_series_sol(LODEstr,opts)

Parameters

 ode - homogeneous linear ODE with polynomial coefficients var - dependent variable, for example y(x) opts - optional arguments of the form keyword=value LODEstr - LODEstruct data structure

Description

 • The dAlembertian_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for the given homogeneous linear ordinary differential equation with polynomial coefficients.
 • If ode is an expression, then it is equated to zero.
 • The routine returns an error message if the differential equation ode does not satisfy the following conditions.
 – ode must be homogeneous and linear in var
 – ode must have polynomial coefficients in the independent variable of var, for example, $x$
 – The coefficients of ode must be either rational numbers or depend rationally on one or more parameters.
 • A homogeneous linear ordinary differential equation with coefficients that are polynomials in $x$ has a linear space of formal power series solutions ${\sum }_{n=0}^{\mathrm{\infty }}v\left(n\right){P}_{n}\left(x\right)$ where $P[n]\left(x\right)$ is one of ${\left(x-a\right)}^{n}$, $\frac{{\left(x-a\right)}^{n}}{n!}$, $\frac{1}{{x}^{n}}$, or $\frac{1}{{x}^{n}n!}$, $a$ is the expansion point, and the sequence $v\left(n\right)$ satisfies a homogeneous linear recurrence.
 • The routine selects such formal power series solutions where $v\left(n\right)$ is a d'Alembertian sequence, that is, $v\left(n\right)$ is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).
 • The routine determines an integer $N\ge 0$ such that $v\left(n\right)$ can be represented in the form of a d'Alembertian term:

$v\left(n\right)=h[1]\left(n\right)\left({\sum }_{{n}_{1}=N}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}h[2]\left({n}_{1}\right)\left({\sum }_{{n}_{2}=N}^{{n}_{1}-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\mathrm{...}\left({\sum }_{{n}_{s}=N}^{{n}_{s-1}-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}h[s+1]\left({n}_{s}\right)\right)\right)\right)\mathrm{\left( + \right)}$

 for all $n\ge N$, where $h[i]\left(n\right)$, $1\le i\le s+1$, is a hypergeometric term (see SumTools[Hypergeometric]):

$h[i]\left(n\right)=h\left(N\right)\left({\prod }_{k=N}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}R\left(k\right)\right)\mathrm{\left( ++ \right)}$

 such that $R\left(k\right)=\frac{h[i]\left(k+1\right)}{h[i]\left(k\right)}$ is rational in $k$ for all $k\ge N$.

Options

 • x=a or 'point'=a
 Specifies the expansion point a. The default is $a=0$. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$.
 • 'free'=C
 Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation if the expansion point is singular.
 • 'indices'=[n,k]
 Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).
 • 'outputHGT'=name
 Specifies the form of representation of hypergeometric terms.  The default value is 'inert'.
 – 'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for ${\prod }_{k=N}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}1$, which is simplified to $1$.
 – 'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).
 – 'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).
 • 'outputDAT'=name
 Specifies the form of representation of the sums in ( + ). The default is 'inert'.
 – 'inert' - the sums are in the inert form, except for trivial sums of the form ${\sum }_{k=u}^{v-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}1$, which are simplified to $v-u$.
 – 'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left({x}^{2}+x-2\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+\left({x}^{2}-x\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-\left(6{x}^{2}+7x\right)y\left(x\right)$
 ${\mathrm{ode}}{:=}\left({{x}}^{{2}}{+}{x}{-}{2}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){-}\left({6}{}{{x}}^{{2}}{+}{7}{}{x}\right){}{y}{}\left({x}\right)$ (1)
 > $\mathrm{dAlembertian_series_sol}\left(\mathrm{ode},y\left(x\right),'\mathrm{outputHGT}'='\mathrm{active}','\mathrm{indices}'=\left[n,k\right]\right)$
 $\left\{\frac{{1}}{{2}}{}{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{{2}}^{{n}}{}{\left({x}{-}{1}\right)}^{{n}}}{{\mathrm{Γ}}{}\left({n}\right)}\right){,}\left(\frac{{315}}{{8}}{}{\sum }_{{n}{=}{7}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{-}\frac{{4095}}{{4}}{-}\frac{{1575}}{{4}}{}{x}{-}{315}{}{\left({x}{+}{2}\right)}^{{2}}{-}\frac{{315}}{{2}}{}{\left({x}{+}{2}\right)}^{{3}}{-}\frac{{105}}{{2}}{}{\left({x}{+}{2}\right)}^{{4}}{-}\frac{{21}}{{2}}{}{\left({x}{+}{2}\right)}^{{5}}\right){}{{\mathrm{_C}}}_{{0}}{+}\left({-}{17143}{}\left({\sum }_{{n}{=}{7}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}\left({\sum }_{{\mathrm{n1}}{=}{7}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left(\frac{{1}}{{6}}\right)}^{{\mathrm{n1}}}{}{\mathrm{Γ}}{}\left({\mathrm{n1}}{+}{1}\right)}{\left({\mathrm{n1}}{-}{5}\right){}\left({\mathrm{n1}}{-}{6}\right)}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{Γ}}{}\left({n}{+}{1}\right)}\right){-}\frac{{40}}{{9}}{}{\sum }_{{n}{=}{7}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}\left({\sum }_{{\mathrm{n1}}{=}{7}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left(\frac{{1}}{{6}}\right)}^{{\mathrm{n1}}}{}{\mathrm{Γ}}{}\left({\mathrm{n1}}{+}{1}\right){}\left({\sum }_{{\mathrm{n2}}{=}{7}}^{{\mathrm{n1}}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({\mathrm{n2}}{-}{5}\right){}{\left({-}{9}\right)}^{{\mathrm{n2}}}}{{\mathrm{Γ}}{}\left({\mathrm{n2}}\right)}\right)}{\left({\mathrm{n1}}{-}{5}\right){}\left({\mathrm{n1}}{-}{6}\right)}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{+}\frac{{997295}}{{486}}{+}\frac{{383575}}{{486}}{}{x}{+}\frac{{153430}}{{243}}{}{\left({x}{+}{2}\right)}^{{2}}{+}\frac{{77435}}{{243}}{}{\left({x}{+}{2}\right)}^{{3}}{+}\frac{{73295}}{{729}}{}{\left({x}{+}{2}\right)}^{{4}}{+}\frac{{18835}}{{729}}{}{\left({x}{+}{2}\right)}^{{5}}{-}\frac{{2540}}{{729}}{}{\left({x}{+}{2}\right)}^{{6}}\right){}{{\mathrm{_C}}}_{{1}}{,}\left({\sum }_{{n}{=}{9}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{512}}{}\frac{\left({n}{-}{8}\right){}{{2}}^{{n}}{}{\left({x}{+}{3}\right)}^{{n}}}{{n}{!}}{-}\frac{{25}}{{256}}{-}\frac{{7}}{{256}}{}{x}{-}\frac{{3}}{{128}}{}{\left({x}{+}{3}\right)}^{{2}}{-}\frac{{5}}{{384}}{}{\left({x}{+}{3}\right)}^{{3}}{-}\frac{{1}}{{192}}{}{\left({x}{+}{3}\right)}^{{4}}{-}\frac{{1}}{{640}}{}{\left({x}{+}{3}\right)}^{{5}}{-}\frac{{1}}{{2880}}{}{\left({x}{+}{3}\right)}^{{6}}{-}\frac{{1}}{{20160}}{}{\left({x}{+}{3}\right)}^{{7}}\right){}{{\mathrm{_C}}}_{{0}}{+}\left({-}\frac{{528299}}{{2198016}}{}{\sum }_{{n}{=}{9}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1024}}{{2835}}{}\frac{\left({n}{-}{8}\right){}{{2}}^{{n}}{}\left({\sum }_{{\mathrm{n1}}{=}{9}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left(\frac{{1}}{{8}}\right)}^{{\mathrm{n1}}}{}{\mathrm{Γ}}{}\left({\mathrm{n1}}{+}{1}\right)}{\left({\mathrm{n1}}{-}{7}\right){}\left({-}{8}{+}{\mathrm{n1}}\right)}\right){}{\left({x}{+}{3}\right)}^{{n}}}{{n}{!}}{+}{\sum }_{{n}{=}{9}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({-}\frac{{2}}{{19683}}{}\frac{\left({n}{-}{8}\right){}{{2}}^{{n}}{}\left({\sum }_{{\mathrm{n1}}{=}{9}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left(\frac{{1}}{{8}}\right)}^{{\mathrm{n1}}}{}{\mathrm{Γ}}{}\left({\mathrm{n1}}{+}{1}\right){}\left({\sum }_{{\mathrm{n2}}{=}{9}}^{{\mathrm{n1}}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({\mathrm{n2}}{-}{7}\right){}{\left({-}{12}\right)}^{{\mathrm{n2}}}{}\left({I}{}\sqrt{{11}}{-}{2}{}{\mathrm{n2}}{-}{7}\right){}\left({I}{}\sqrt{{11}}{+}{2}{}{\mathrm{n2}}{+}{7}\right)}{\left({\mathrm{n2}}{+}{1}\right){}{\mathrm{Γ}}{}\left({\mathrm{n2}}{+}{1}\right)}\right)}{\left({\mathrm{n1}}{-}{7}\right){}\left({-}{8}{+}{\mathrm{n1}}\right)}\right){}{\left({x}{+}{3}\right)}^{{n}}}{\left({I}{}\sqrt{{11}}{-}{25}\right){}\left({I}{}\sqrt{{11}}{+}{25}\right){}{n}{!}}\right){+}\frac{{869574607}}{{273468358656}}{+}\frac{{243522833}}{{273468358656}}{}{x}{+}\frac{{3853999}}{{5064228864}}{}{\left({x}{+}{3}\right)}^{{2}}{+}\frac{{366367}}{{859963392}}{}{\left({x}{+}{3}\right)}^{{3}}{+}\frac{{1269989}}{{7596343296}}{}{\left({x}{+}{3}\right)}^{{4}}{+}\frac{{11983373}}{{227890298880}}{}{\left({x}{+}{3}\right)}^{{5}}{+}\frac{{1166693}}{{113945149440}}{}{\left({x}{+}{3}\right)}^{{6}}{+}\frac{{572503}}{{265872015360}}{}{\left({x}{+}{3}\right)}^{{7}}{-}\frac{{93781}}{{398808023040}}{}{\left({x}{+}{3}\right)}^{{8}}\right){}{{\mathrm{_C}}}_{{1}}\right\}$ (2)