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):

for all $n\ge N$.

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[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.

'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).

$\mathrm{with}\left(\mathrm{Slode}\right)\:$

$\mathrm{ode}≔2x\left(x-1\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+\left(7x-3\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

$\mathrm{ode}≔2x\left(x-1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(7x-3\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2y\left(x\right)=0$

$\mathrm{hypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,x=-1\right)$

$\frac{{\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{\_n}\right){\left(x+1\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{\sqrt{\mathrm{\pi }}}$

$\mathrm{hypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,x=0\right)$

${\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}+1\right)x^{\mathrm{\_n}}}{2\mathrm{\_n}+1}\right)$

$\mathrm{hypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left\{{\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}+1\right)x^{\mathrm{\_n}}}{2\mathrm{\_n}+1}\right)\,\frac{{\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{\_n}\right){\left(x+1\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{\sqrt{\mathrm{\pi }}}\,\frac{{\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{\_n}\right){\left(\mathrm{-1}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{\mathrm{\_n}}}{\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)}\right)}{\sqrt{\mathrm{\pi }}}\right\}$

$\mathrm{ode1}≔\left(\frac{81}{4}{x}^3-3x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+\left(\frac{567}{4}{x}^2-\frac{39}{2}x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(207x-\frac{45}{2}\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+45y\left(x\right)=\frac{\frac{3}{2}\left(5x^4+330x^3+1137x^2-32x-60\right)}{{\left(x-1\right)}^6}$

$\mathrm{ode1}≔\left(\frac{81}{4}{x}^3-3x^2\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+\left(\frac{567}{4}{x}^2-\frac{39}{2}x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(207x-\frac{45}{2}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+45y\left(x\right)=\frac{3\left(5x^4+330x^3+1137x^2-32x-60\right)}{2{\left(x-1\right)}^6}$

$\mathrm{hypergeom\_series\_sol}\left(\mathrm{ode1}\,y\left(x\right)\,x=0\,'\mathrm{indices}'=\left[n\,k\right]\right)$

$\frac{54{\mathrm{\_C}}_0\sqrt{3}\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+\frac{5}{3}\right)\mathrm{\Gamma }\left(n+\frac{4}{3}\right){27}^n{x}^n}{\mathrm{\Gamma }\left(2n+5\right)}\right)}{\mathrm{\pi }}-\frac{9\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\left(4n+7\right)x^n}{\left(n+1\right)\left(n+2\right)}\right)}{4}+\frac{\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\left(-2n+\sqrt{13}\right)\left(2n+\sqrt{13}\right)x^n}{\left(n+1\right)\left(n+2\right)}\right)}{4}+\frac{75\left(\sum _{n=0}^{\mathrm{\infty }}\left(\prod _{k=0}^{n-1}\frac{\left(4k^3+12k^2+12k+29\right)\left(k+1\right)}{\left(4k^3+25\right)\left(k+3\right)}\right)x^n\right)}{8}+\frac{\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\left(\mathrm{I}\sqrt{14}-2n\right)\left(\mathrm{I}\sqrt{14}+2n\right)\left(-2n+\sqrt{14}\right)\left(2n+\sqrt{14}\right)x^n}{\left(n+1\right)\left(n+2\right)}\right)}{16}$