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 hypergeom_formal_sol
 formal solutions with hypergeometric series coefficients for a linear ODE

 Calling Sequence hypergeom_formal_sol(ode, var, opts) hypergeom_formal_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 hypergeom_formal_sol command returns formal solutions with hypergeometric series 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 basis of formal solutions (see DEtools[formal_sol]). A formal solution contains a finite number of power series ${\sum }_{n=0}^{\mathrm{\infty }}v\left(n\right){T}^{n}$ where $T$ is a parameter and the sequence $v\left(n\right)$ satisfies a linear recurrence (homogeneous or inhomogeneous).
 • This routine selects solutions that contain series 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 routine 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):

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

 for all $n\ge N$.

Options

 • 'parameter'=T
 Specifies the name T that is used to denote $a{x}^{\frac{1}{r}}$ where $a$ is a constant and $r$ is called the ramification index. If this option is given, then the routine expresses the formal solutions in terms of T and returns a list of lists each of which is of the form [formal solution, relation between T and x]. Otherwise, it returns the formal solutions in terms of ${x}^{\frac{1}{r}}$.
 • 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 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

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔3xy\left(x\right)+\left(4+4x\right)x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left(-3-3x\right){x}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)$
 ${\mathrm{ode}}{:=}{3}{}{x}{}{y}{}\left({x}\right){+}\left({4}{+}{4}{}{x}\right){}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}\left({-}{3}{-}{3}{}{x}\right){}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right)$ (1)
 > $\mathrm{hypergeom_formal_sol}\left(\mathrm{ode},y\left(x\right)\right)$
 $\frac{{9}}{{4}}{}\frac{{{\mathrm{_C}}}_{{0}}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\mathrm{Γ}}{}\left({\mathrm{_n}}{-}\frac{{7}}{{6}}{-}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{-}\frac{{7}}{{6}}{+}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{{x}}^{{\mathrm{_n}}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{-}\frac{{4}}{{3}}\right)}\right)}{{\mathrm{Γ}}{}\left({-}\frac{{7}}{{6}}{-}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{\mathrm{Γ}}{}\left({-}\frac{{7}}{{6}}{+}\frac{{1}}{{6}}{}\sqrt{{85}}\right)}{+}\frac{{56}}{{81}}{}\frac{{{x}}^{{7}{/}{3}}{}{{\mathrm{_C}}}_{{1}}{}{\mathrm{π}}{}\sqrt{{3}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{7}}{{6}}{-}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{7}}{{6}}{+}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{{x}}^{{\mathrm{_n}}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{10}}{{3}}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{{\mathrm{Γ}}{}\left(\frac{{7}}{{6}}{-}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{\mathrm{Γ}}{}\left(\frac{{7}}{{6}}{+}\frac{{1}}{{6}}{}\sqrt{{85}}\right){}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}$ (2)
 > $\mathrm{hypergeom_formal_sol}\left(\mathrm{ode},y\left(x\right),'\mathrm{parameter}'=t,'\mathrm{outputHGT}'='\mathrm{inert}'\right)$
 $\left[\left[{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{3}{}{{\mathrm{_k}}}^{{2}}{-}{7}{}{\mathrm{_k}}{-}{3}}{{3}{}{{\mathrm{_k}}}^{{2}}{-}{\mathrm{_k}}{-}{4}}\right){}{{t}}^{{\mathrm{_n}}}\right){,}{x}{=}{t}\right]{,}\left[{{t}}^{{7}{/}{3}}{}{{\mathrm{_C}}}_{{1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{3}{}{{\mathrm{_k}}}^{{2}}{+}{7}{}{\mathrm{_k}}{-}{3}}{{3}{}{{\mathrm{_k}}}^{{2}}{+}{13}{}{\mathrm{_k}}{+}{10}}\right){}{{t}}^{{\mathrm{_n}}}\right){,}{x}{=}{t}\right]\right]$ (3)