Slode[hypergeom_formal_sol] - formal solutions with hypergeometric series coefficients for a linear ODE
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Calling Sequence
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hypergeom_formal_sol(ode, var, opts)
hypergeom_formal_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 hypergeom_formal_sol command returns formal solutions with hypergeometric series 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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for all .
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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 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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![(9/4)*_C[0]*GAMMA(2/3)*(Sum((-1)^_n*GAMMA(_n-7/6-(1/6)*85^(1/2))*GAMMA(_n-7/6+(1/6)*85^(1/2))*x^_n/(GAMMA(_n+1)*GAMMA(_n-4/3)), _n = 0 .. infinity))/(GAMMA(-7/6-(1/6)*85^(1/2))*GAMMA(-7/6+(1/6)*85^(1/2)))+(56/81)*x^(7/3)*_C[1]*Pi*3^(1/2)*(Sum((-1)^_n*GAMMA(_n+7/6-(1/6)*85^(1/2))*GAMMA(_n+7/6+(1/6)*85^(1/2))*x^_n/(GAMMA(_n+10/3)*GAMMA(_n+1)), _n = 0 .. infinity))/(GAMMA(7/6-(1/6)*85^(1/2))*GAMMA(7/6+(1/6)*85^(1/2))*GAMMA(2/3))](/support/helpjp/helpview.aspx?si=7115/file02777/math270.png)
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![[[_C[0]*(Sum((-1)^_n*(Product((-7*_k-3+3*_k^2)/(3*_k^2-_k-4), _k = 0 .. _n-1))*t^_n, _n = 0 .. infinity)), x = t], [t^(7/3)*_C[1]*(Sum((-1)^_n*(Product((7*_k-3+3*_k^2)/(3*_k^2+13*_k+10), _k = 0 .. _n-1))*t^_n, _n = 0 .. infinity)), x = t]]](/support/helpjp/helpview.aspx?si=7115/file02777/math277.png)
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