formal m-sparse m-hypergeometric power series solutions for a linear ODE - Maple Help

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Slode[mhypergeom_series_sol] - formal m-sparse m-hypergeometric power series solutions for a linear ODE

Calling Sequence

mhypergeom_series_sol(ode, var, opts)

mhypergeom_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 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, 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 linear space of formal power series solutions n=0vnPnx where Pnx is one of xan, xann!, 1xn, or 1xnn!, a is the expansion point, and the sequence vn satisfies a homogeneous linear recurrence.

• 

This routine selects such formal power series solutions where for an integer m2 there is an integer i such that

– 

vn0 only if nimodm=0, and

– 

vn+1m+i=pnvmn+i for all sufficiently large n, where pn is a rational function.

• 

The routine determines an integer N0 such that the elements vmn+i can be represented in the form of hypergeometric term (see SumTools[Hypergeometric], LREtools):

vmn+i=vNm+ik=Nn1pkm+i ( * )

  

for all nN.

Options

• 

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 .

  

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 m=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 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 k=Nn11, 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

withSlode:

ode:=ⅆ2ⅆx2yx+x1yx

ode:=ⅆ2ⅆx2yx+x1yx

(1)

mhypergeom_series_solode,yx

_C0Γ23_n=0∞19_nx13_nΓ_n+1Γ_n+23,29_C0π3_n=0∞19_nx13_n+1Γ_n+43Γ_n+1Γ23

(2)

mhypergeom_series_solode,yx,'indices'=n,k,'outputHGT'='inert'

_C0n=0∞13nk=0n113k2+5k+2x13n,_C0n=0∞13nk=0n113k2+7k+4x13n+1

(3)

Inhomogeneous equations are handled:

ode1:=z2ⅆ2ⅆz2yz+3zⅆⅆzyz+z215yz=1

ode1:=z2ⅆ2ⅆz2yz+3zⅆⅆzyz+z215yz=1

(4)

mhypergeom_series_solode1,yz,z=∞,'sparseorder'=2,'outputHGT'='inert'

115_n=0∞1_n_k=0_n14_k24_k15z2_n115

(5)

See Also

LODEstruct, Slode, Slode[candidate_mpoints], Slode[hypergeom_series_sol], Slode[msparse_series_sol]


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