gfun[listtodiffeq] - find a linear differential equation for the generating function
gfun[seriestodiffeq] - find a linear differential equation satisfied by a series
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Calling Sequence
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listtodiffeq(l, y(x), [typelist])
seriestodiffeq(s, y(x), [typelist])
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Parameters
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l
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list
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y
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name; function name
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x
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name; variable of the function y
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typelist
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(optional) list of generating function types. The default is 'ogf','egf'. For a complete list of types, see gftypes.
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s
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series
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Description
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The listtodiffeq(l, y(x), [typelist]) command computes a linear differential equation in y(x) with polynomial coefficients in x satisfied by the generating function y(x) of the expressions in l. This generating function is one of the types specified by typelist, for example, ordinary (ogf) or exponential (egf). For a complete list of available generating function types, see gftypes.
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The seriestodiffeq(l, y(x), [typelist]) command computes a linear differential equation in y(x) with polynomial coefficients in x satisfied by the generating function y(x) of the expressions in s. This generating function is one of the types specified by typelist, for example, ordinary (ogf) or exponential (egf). For a complete list of available generating function types, see gftypes.
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If typelist contains more than one element, these types are considered in the order that they are listed.
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If typelist is not specified, the default typelist, 'ogf','egf', is used. The function returns a list whose first element is the differential equation satisfied by the generating function. The second element is the generating function type for which an equation was found.
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In the implementation, the maximal order is 2 and the maximum degree of the coefficients is 3. You can change these degree specifications by modifying the variables gfun['maxordereqn'] and gfun['maxdegcoeff'].
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If sufficiently many terms were specified and no solution is found, then the generating function does not satisfy any linear differential equation of order less than or equal to gfun['maxordereqn'] with coefficients of degree less than or equal to gfun['maxdegcoeff'].
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Examples
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![l := [1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590]](/support/helpjp/helpview.aspx?si=6508/file04158/math166.png)
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![[{-(-4+27*x)*x^2*(diff(y(x), `$`(x, 2)))+(18*x-108*x^2)*(diff(y(x), x))-12+(12-60*x)*y(x), y(0) = 1, (D(y))(0) = 2}, ogf]](/support/helpjp/helpview.aspx?si=6508/file04158/math173.png)
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