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'].

$\mathrm{with}\left(\mathrm{gfun}\right)\:$

$l≔\left[1\,2\,6\,22\,91\,408\,1938\,9614\,49335\,260130\,1402440\,7702632\,42975796\,243035536\,1390594458\,8038677054\,46892282815\,275750636070\,1633292229030\,9737153323590\right]\:$

$\mathrm{listtodiffeq}\left(l\,y\left(x\right)\right)$

$\left[\left\{-\left(27x-4\right)x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(-108x^2+18x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-12+\left(-60x+12\right)y\left(x\right)\,y\left(0\right)=1\,\mathrm{D}\left(y\right)\left(0\right)=2\right\}\,\mathrm{ogf}\right]$

$s≔\mathrm{series}\left(\frac{\exp \left(x\right)}{\mathrm{sqrt}\left(1-x\right)}\,x\,7\right)$

$s≔1+\frac{3}{2}x+\frac{11}{8}{x}^2+\frac{53}{48}{x}^3+\frac{115}{128}{x}^4+\frac{2947}{3840}{x}^5+\frac{31411}{46080}{x}^6+O\left(x^7\right)$

$\mathrm{seriestodiffeq}\left(s\,y\left(x\right)\right)$

$\left[\left\{\left(-2x+2\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(2x-3\right)y\left(x\right)\,y\left(0\right)=1\right\}\,\mathrm{ogf}\right]$