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gfun

 listtodiffeq
 find a linear differential equation for the generating function
 seriestodiffeq
 find a linear differential equation satisfied by a series

 Calling Sequence listtodiffeq(l, y(x), [typelist]) seriestodiffeq(s, y(x), [typelist])

Parameters

 l - list y - name; function name x - name; variable of the function y typelist - (optional) list of generating function types. The default is 'ogf','egf'. For a complete list of types, see gftypes. s - series

Description

 • 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.
 • 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.
 • If typelist contains more than one element, these types are considered in the order that they are listed.
 • 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.
 • 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'].
 • 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'].

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $l≔\left[1,2,6,22,91,408,1938,9614,49335,260130,1402440,7702632,42975796,243035536,1390594458,8038677054,46892282815,275750636070,1633292229030,9737153323590\right]:$
 > $\mathrm{listtodiffeq}\left(l,y\left(x\right)\right)$
 $\left[\left\{{-}\left({27}{}{x}{-}{4}\right){}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{108}{}{{x}}^{{2}}{+}{18}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{12}{+}\left({-}{60}{}{x}{+}{12}\right){}{y}{}\left({x}\right){,}{y}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{2}\right\}{,}{\mathrm{ogf}}\right]$ (1)
 > $s≔\mathrm{series}\left(\frac{\mathrm{exp}\left(x\right)}{\mathrm{sqrt}\left(1-x\right)},x,7\right)$
 ${s}{≔}{1}{+}\frac{{3}}{{2}}{}{x}{+}\frac{{11}}{{8}}{}{{x}}^{{2}}{+}\frac{{53}}{{48}}{}{{x}}^{{3}}{+}\frac{{115}}{{128}}{}{{x}}^{{4}}{+}\frac{{2947}}{{3840}}{}{{x}}^{{5}}{+}\frac{{31411}}{{46080}}{}{{x}}^{{6}}{+}{O}{}\left({{x}}^{{7}}\right)$ (2)
 > $\mathrm{seriestodiffeq}\left(s,y\left(x\right)\right)$
 $\left[\left\{\left({-}{2}{}{x}{+}{2}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({2}{}{x}{-}{3}\right){}{y}{}\left({x}\right){,}{y}{}\left({0}\right){=}{1}\right\}{,}{\mathrm{ogf}}\right]$ (3)

 See Also