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,

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  has a linear space of formal power series solutions  where  is one of , , , or ,  is the expansion point, and the sequence  satisfies a homogeneous linear recurrence.

This routine selects such formal power series solutions where for an integer  there is an integer  such that

 only if , and

 for all sufficiently large , where  is a rational function.

The routine determines an integer  such that the elements  can be represented in the form of hypergeometric term (see SumTools[Hypergeometric], LREtools):

for all .

x=a or 'point'=a

Specifies the expansion point a. The default is . It can be an algebraic number, depending rationally on some parameters, or .

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  only. Otherwise, it returns a set of m-sparse m-hypergeometric power series solution for all possible values of .

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 . 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 , which is simplified to .

'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).