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 hypergeom_series_sol
 formal power series solutions with hypergeometric coefficients for a linear ODE Calling Sequence hypergeom_series_sol(ode, var,opts) hypergeom_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 hypergeom_series_sol command returns one formal power series solution or a set of formal power series solutions of the given linear ordinary differential equation with polynomial coefficients. The ODE must be either homogeneous or inhomogeneous with a right-hand side that is a polynomial, a rational function, or a "nice" power series (see LODEstruct) in the independent variable $x$.
 • If ode is an expression, then it is equated to zero.
 • The command returns an error message if the differential equation ode does not satisfy the following conditions.
 – ode must be linear in var
 – ode must be homogeneous or have a right-hand side that is rational or a "nice" power series in $x$
 – The coefficients of ode must be polynomial in the independent variable of var, for example, $x$, over the rational number field which can be extended by 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. In the case of an inhomogeneous equation with a right-hand side that is a "nice" power series, $v\left(n\right)$ satisfies an inhomogeneous linear recurrence.
 • The command selects such formal power series solutions where $v\left(n+1\right)=p\left(n\right)v\left(n\right)$ for all sufficiently large $n$, where $p\left(n\right)$ is a rational function.
 • This command determines an integer $N\ge 0$ such that $v\left(n\right)$ can be represented in the form of hypergeometric term (see SumTools[Hypergeometric],LREtools):

$v\left(n\right)=v\left(N\right)\left({\prod }_{k=N}^{n-1}p\left(k\right)\right)\mathrm{\left( * \right)}$

 for all $n\ge N$. Options

 • x=a or 'point'=a
 Specifies the expansion point in the case of a homogeneous equation or an inhomogeneous equation with rational right-hand side. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$. In the case of a "nice" series right-hand side the expansion point is given by the right-hand side and cannot be changed.
 If this option is given, then the command returns one formal power series solution at a with hypergeometric coefficients if it exists; otherwise, it returns NULL. If a is not given, it returns a set of formal power series solutions with hypergeometric coefficients for all possible points that are determined by Slode[candidate_points](ode,var,'type'='hypergeometric').
 • 'free'=C
 Specifies a base name C to use for free variables C, C, 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.
 • '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 ${\prod }_{k=N}^{n-1}1$, 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

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔2x\left(x-1\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(7x-3\right)\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode}}{≔}{2}{}{x}{}\left({x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({7}{}{x}{-}{3}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{hypergeom_series_sol}\left(\mathrm{ode},y\left(x\right),x=-1\right)$
 $\frac{{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({x}{+}{1}\right)}^{{\mathrm{_n}}}}{{\mathrm{_n}}{!}}\right)}{\sqrt{{\mathrm{\pi }}}}$ (2)
 > $\mathrm{hypergeom_series_sol}\left(\mathrm{ode},y\left(x\right),x=0\right)$
 ${{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right)$ (3)
 > $\mathrm{hypergeom_series_sol}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left\{{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right){,}{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({x}{+}{1}\right)}^{{\mathrm{_n}}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{_n}}{!}}\right){,}\frac{{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({-1}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{\mathrm{_n}}}}{{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{\sqrt{{\mathrm{\pi }}}}\right\}$ (4)

Inhomogeneous equations are handled:

 > $\mathrm{ode1}≔\left(\frac{81}{4}{x}^{3}-3{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x,x,x\right)+\left(\frac{567}{4}{x}^{2}-\frac{39}{2}x\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(207x-\frac{45}{2}\right)\mathrm{diff}\left(y\left(x\right),x\right)+45y\left(x\right)=\frac{\frac{3}{2}\left(5{x}^{4}+330{x}^{3}+1137{x}^{2}-32x-60\right)}{{\left(x-1\right)}^{6}}$
 ${\mathrm{ode1}}{≔}\left(\frac{{81}}{{4}}{}{{x}}^{{3}}{-}{3}{}{{x}}^{{2}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left(\frac{{567}}{{4}}{}{{x}}^{{2}}{-}\frac{{39}}{{2}}{}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({207}{}{x}{-}\frac{{45}}{{2}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{45}{}{y}{}\left({x}\right){=}\frac{{3}{}\left({5}{}{{x}}^{{4}}{+}{330}{}{{x}}^{{3}}{+}{1137}{}{{x}}^{{2}}{-}{32}{}{x}{-}{60}\right)}{{2}{}{\left({x}{-}{1}\right)}^{{6}}}$ (5)
 > $\mathrm{hypergeom_series_sol}\left(\mathrm{ode1},y\left(x\right),x=0,'\mathrm{indices}'=\left[n,k\right]\right)$
 $\frac{{54}{}{{\mathrm{_C}}}_{{0}}{}\sqrt{{3}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left({n}{+}\frac{{5}}{{3}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{4}}{{3}}\right){}{{27}}^{{n}}{}{{x}}^{{n}}}{{\mathrm{\Gamma }}{}\left({2}{}{n}{+}{5}\right)}\right)}{{\mathrm{\pi }}}{-}\frac{{9}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({4}{}{n}{+}{7}\right){}{{x}}^{{n}}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}\right)}{{4}}{+}\frac{\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({-}{2}{}{n}{+}\sqrt{{13}}\right){}\left({2}{}{n}{+}\sqrt{{13}}\right){}{{x}}^{{n}}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}\right)}{{4}}{+}\frac{{75}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({4}{}{{k}}^{{3}}{+}{12}{}{{k}}^{{2}}{+}{12}{}{k}{+}{29}\right){}\left({k}{+}{1}\right)}{\left({4}{}{{k}}^{{3}}{+}{25}\right){}\left({k}{+}{3}\right)}\right){}{{x}}^{{n}}\right)}{{8}}{+}\frac{\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({I}{}\sqrt{{14}}{-}{2}{}{n}\right){}\left({I}{}\sqrt{{14}}{+}{2}{}{n}\right){}\left({-}{2}{}{n}{+}\sqrt{{14}}\right){}\left({2}{}{n}{+}\sqrt{{14}}\right){}{{x}}^{{n}}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}\right)}{{16}}$ (6)

 See Also