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Slode[dAlembertian_series_sol] - formal power series solutions with d'Alembertian coefficients for a linear ODE
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Calling Sequence
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dAlembertian_series_sol(ode,var,opts)
dAlembertian_series_sol(LODEstr,opts)
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Parameters
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ode
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homogeneous linear ODE with polynomial coefficients
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var
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dependent variable, for example y(x)
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opts
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optional arguments of the form keyword=value
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LODEstr
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LODEstruct data structure
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Description
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The dAlembertian_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for the given homogeneous linear ordinary differential equation with polynomial coefficients.
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If ode is an expression, then it is equated to zero.
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The routine returns an error message if the differential equation ode does not satisfy the following conditions.
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ode must be homogeneous and linear in var
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ode must have polynomial coefficients in the independent variable of var, for example, 
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The coefficients of ode must be either rational numbers or depend rationally on one or more parameters.
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The routine selects such formal power series solutions where  is a d'Alembertian sequence, that is,  is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).
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The routine determines an integer  such that  can be represented in the form of a d'Alembertian term:
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*(Sum(h[2](n[1])*(Sum(`...`*(Sum(h[s+1](n[s]), n[s] = N .. n[s-1]-1)), n[2] = N .. n[1]-1)), n[1] = N .. n-1))*` ( + )`](/support/helpjp/helpview.aspx?si=7493/file02776/math144.png)
 = h(N)*(Product(R(k)*` ( ++ )`, k = N .. n-1))](/support/helpjp/helpview.aspx?si=7493/file02776/math160.png)
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Options
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Specifies the expansion point a. The default is  . It can be an algebraic number, depending rationally on some parameters, or  .
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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.
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Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).
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Specifies the form of representation of hypergeometric terms. The default value is 'inert'.
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'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for  , which is simplified to  .
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'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).
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'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).
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Specifies the form of representation of the sums in ( + ). The default is 'inert'.
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'inert' - the sums are in the inert form, except for trivial sums of the form  , which are simplified to  .
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'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.
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