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 command returns an error message if the differential equation ode does not satisfy the following conditions.

ode must be homogeneous and linear in var

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.

This command 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 $n-i\mathbf{mod}m=0$, and

$v\left(\left(n+1\right)m+i\right)=p\left(n\right)v\left(nm+i\right)$ for all sufficiently large $n$, where $p\left(n\right)$ is a rational function.

The command determines an integer $N\ge 0$ such that the elements $v\left(nm+i\right)$ 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 a. 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 command 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.

'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).

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

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(x-1\right)y\left(x\right)$

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

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

$\left\{{\mathrm{\_C}}_0\mathrm{\Gamma }\left(\frac{2}{3}\right)\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{9}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{3\mathrm{\_n}}}{\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)\mathrm{\Gamma }\left(\mathrm{\_n}+\frac{2}{3}\right)}\right)\,\frac{2{\mathrm{\_C}}_0\mathrm{\pi }\sqrt{3}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{9}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{3\mathrm{\_n}+1}}{\mathrm{\Gamma }\left(\mathrm{\_n}+\frac{4}{3}\right)\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)}\right)}{9\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right\}$

$\mathrm{mhypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{indices}'=\left[n\,k\right]\,'\mathrm{outputHGT}'='\mathrm{inert}'\right)$

$\left\{{\mathrm{\_C}}_0\left(\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{3}\right)}^n{\left(x-1\right)}^{3n}}{\prod _{k=0}^{n-1}\left(3k^2+5k+2\right)}\right)\,{\mathrm{\_C}}_0\left(\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{3}\right)}^n{\left(x-1\right)}^{3n+1}}{\prod _{k=0}^{n-1}\left(3k^2+7k+4\right)}\right)\right\}$

$\mathrm{ode1}≔z^2\mathrm{diff}\left(y\left(z\right)\,z\,z\right)+3z\mathrm{diff}\left(y\left(z\right)\,z\right)+\left(z^2-15\right)y\left(z\right)=1$

$\mathrm{ode1}≔z^2\left(\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)\right)+3z\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)+\left(z^2-15\right)y\left(z\right)=1$

$\mathrm{mhypergeom\_series\_sol}\left(\mathrm{ode1}\,y\left(z\right)\,z=\mathrm{\infty }\,'\mathrm{sparseorder}'=2\,'\mathrm{outputHGT}'='\mathrm{inert}'\right)$

$\left\{\frac{\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\_n}}\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n}-1}\left(4{\mathrm{\_k}}^2-4\mathrm{\_k}-15\right)\right)}{z^{2\mathrm{\_n}}}\right)}{15}-\frac{1}{15}\right\}$