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 mhypergeom_series_sol
 formal m-sparse m-hypergeometric power series solutions for a linear ODE

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

Parameters

 ode - 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 mhypergeom_series_sol command returns a set of formal m-sparse m-hypergeometric power series solutions of the given 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.
 • This routine selects such formal power series solutions where for an integer $m\ge 2$ there is an integer $i$ such that
 – $v\left(n\right)\ne 0$ only if $n-i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m=0$, and
 – $v\left(\left(n+1\right)m+i\right)=p\left(n\right)v\left(mn+i\right)$ for all sufficiently large $n$, where $p\left(n\right)$ is a rational function.
 • The routine determines an integer $N\ge 0$ such that the elements $v\left(mn+i\right)$ can be represented in the form of hypergeometric term (see SumTools[Hypergeometric], LREtools):

$v\left(mn+i\right)=v\left(Nm+i\right)\left({\prod }_{k=N}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}p\left(km+i\right)\right)\mathrm{\left( * \right)}$

 for all $n\ge N$.

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 $\mathrm{\infty }$.
 If this option is given, then the command returns a set of m-sparse m-hypergeometric power series solutions at the given point a. Otherwise, it returns a set of m-sparse m-hypergeometric power series solutions for all possible points that are determined by Slode[candidate_mpoints](ode,var).
 • 'sparseorder'=m0
 Specifies an integer m0. If this option is given, then the procedure computes a set of m-sparse m-hypergeometric power series solutions with $m=\mathrm{m0}$ only. Otherwise, it returns a set of m-sparse m-hypergeometric power series solution for all possible values of $m$.
 If both an expansion point and a sparse order are given, then the command can also compute a set of m-sparse m-hypergeometric series solutions for an inhomogeneous equation with polynomial coefficients and a right-hand side that is rational in the independent variable $x$. Otherwise, the equation has to be homogeneous.
 • '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 names for dummy variables. The default values are the global names _n and _k. The name n is used as the summation index in the power series. The name k is used as the product index in ( * ).
 • 'outputHGT'=name
 Specifies the form of representation of hypergeometric terms.  The default value is 'active'.
 – '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).

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+\left(x-1\right)y\left(x\right)$
 ${\mathrm{ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}\left({x}{-}{1}\right){}{y}{}\left({x}\right)$ (1)
 > $\mathrm{mhypergeom_series_sol}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left\{{{\mathrm{_C}}}_{{0}}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{2}}{{3}}\right)}\right){,}\frac{{2}}{{9}}{}\frac{{{\mathrm{_C}}}_{{0}}{}{\mathrm{π}}{}\sqrt{{3}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}{+}{1}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{4}}{{3}}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}\right\}$ (2)
 > $\mathrm{mhypergeom_series_sol}\left(\mathrm{ode},y\left(x\right),'\mathrm{indices}'=\left[n,k\right],'\mathrm{outputHGT}'='\mathrm{inert}'\right)$
 $\left\{{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\left({-}\frac{{1}}{{3}}\right)}^{{n}}{}{\prod }_{{k}{=}{0}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{3}{}{{k}}^{{2}}{+}{5}{}{k}{+}{2}}\right){}{\left({x}{-}{1}\right)}^{{3}{}{n}}\right){,}{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\left({-}\frac{{1}}{{3}}\right)}^{{n}}{}{\prod }_{{k}{=}{0}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{3}{}{{k}}^{{2}}{+}{7}{}{k}{+}{4}}\right){}{\left({x}{-}{1}\right)}^{{3}{}{n}{+}{1}}\right)\right\}$ (3)

Inhomogeneous equations are handled:

 > $\mathrm{ode1}≔{z}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{z}^{2}}y\left(z\right)\right)+3z\left(\frac{ⅆ}{ⅆz}y\left(z\right)\right)+\left({z}^{2}-15\right)y\left(z\right)=1$
 ${\mathrm{ode1}}{:=}{{z}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{y}{}\left({z}\right)\right){+}{3}{}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{y}{}\left({z}\right)\right){+}\left({{z}}^{{2}}{-}{15}\right){}{y}{}\left({z}\right){=}{1}$ (4)
 > $\mathrm{mhypergeom_series_sol}\left(\mathrm{ode1},y\left(z\right),z=\mathrm{∞},'\mathrm{sparseorder}'=2,'\mathrm{outputHGT}'='\mathrm{inert}'\right)$
 $\left\{\frac{{1}}{{15}}{}{\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({4}{}{{\mathrm{_k}}}^{{2}}{-}{4}{}{\mathrm{_k}}{-}{15}\right)}{{{z}}^{{2}{}{\mathrm{_n}}}}{-}\frac{{1}}{{15}}\right\}$ (5)