Slode[hypergeom_series_sol] - formal power series solutions with hypergeometric coefficients for a linear ODE
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Calling Sequence
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hypergeom_series_sol(ode, var,opts)
hypergeom_series_sol(LODEstr,opts)
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Parameters
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ode
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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 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 .
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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 linear in var
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ode must have polynomial coefficients in
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ode must be homogeneous or have a right-hand side that is rational or a "nice" power series in
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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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for all .
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Options
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Specifies the expansion point in the case of a homogeneous equation or an inhomogeneous equation with rational right-hand side. The default is . 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.
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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').
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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 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 ( * ).
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Specifies the form of representation of hypergeometric terms. The default value is 'active'.
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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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Examples
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Inhomogeneous equations are handled:
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![54*_C[0]*3^(1/2)*(Sum(GAMMA(n+4/3)*GAMMA(n+5/3)*27^n*x^n/GAMMA(2*n+5), n = 0 .. infinity))/Pi-(9/4)*(Sum((7+4*n)*x^n/((n+1)*(n+2)), n = 0 .. infinity))+(1/4)*(Sum((-2*n+13^(1/2))*(2*n+13^(1/2))*x^n/((n+1)*(n+2)), n = 0 .. infinity))+(75/8)*(Sum((product((29+4*k^3+12*k^2+12*k)*(k+1)/((k+3)*(25+4*k^3)), k = 0 .. n-1))*x^n, n = 0 .. infinity))+(1/16)*(Sum((-2*n+I*14^(1/2))*(2*n+I*14^(1/2))*(-2*n+14^(1/2))*(2*n+14^(1/2))*x^n/((n+1)*(n+2)), n = 0 .. infinity))](/support/helpjp/helpview.aspx?si=7350/file02778/math304.png)
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