formal power series solutions with hypergeometric coefficients for a linear ODE - Maple Help

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Slode[hypergeom_series_sol] - formal power series solutions with hypergeometric coefficients for a linear ODE

Calling Sequence

hypergeom_series_sol(ode, var,opts)





linear ODE with polynomial coefficients



dependent variable, for example y(x)



optional arguments of the form keyword=value



LODEstruct data structure



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


ode must be linear in var


ode must have polynomial coefficients in x


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 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. In the case of an inhomogeneous equation with a right-hand side that is a "nice" power series, vn satisfies an inhomogeneous linear recurrence.


The routine selects such formal power series solutions where vn+1=pnvn for all sufficiently large n, where pn is a rational function.


This routine determines an integer N0 such that vn can be represented in the form of hypergeometric term (see SumTools[Hypergeometric],LREtools):

vn=vNk=Nn1pk ( * )


for all nN.



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. The default is a=0. It can be an algebraic number, depending rationally on some parameters, or . 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').




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.




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 ( * ).




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).















Inhomogeneous equations are handled:







See Also

LODEstruct, Slode, Slode[candidate_points], Slode[mhypergeom_series_sol], Slode[polynomial_series_sol], Slode[rational_series_sol]

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