listtoalgeq - Maple Help

gfun

 listtoalgeq
 find an algebraic equation satisfied by a generating function
 seriestoalgeq
 find an algebraic equation satisfied by a series

 Calling Sequence listtoalgeq(l, y(x), [typelist]) seriestoalgeq(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 listtoalgeq(l, y(x), [typelist]) command computes a polynomial equation in y and x satisfied by the generating function y(x) of the expressions in l.  The 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 seriestoalgeq(s, y(x), [typelist]) command computes a polynomial equation in y and x satisfied by the generating function y(x) of the expressions in s.  The 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 polynomial in y(x) and x that was found.  The second element is the generating function type to which the first element corresponds.
 • In the implementation, the maximal degree of y is 6 and the maximum degree of the coefficients is 3. You can change these degree specifications by modifying the variables gfun['maxdegeqn'] and gfun['maxdegcoeff'].
 • If sufficiently many terms are specified and no solution is found, then the generating function does not satisfy any algebraic equation of degree less than or equal to gfun['maxdegeqn'] with coefficients of degree less than or equal to gfun['maxdegcoeff'].

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $l≔\left[1,1,2,5,14,42,132,429,1430,4862,16796,58786\right]:$
 > $\mathrm{listtoalgeq}\left(l,y\left(x\right)\right)$
 $\left[{-}{1}{+}{y}{}\left({x}\right){-}{x}{}{{y}{}\left({x}\right)}^{{2}}{,}{\mathrm{ogf}}\right]$ (1)
 > $s≔\mathrm{series}\left(1-\mathrm{sqrt}\left(1-4x\right),x,9\right)$
 ${s}{≔}{2}{}{x}{+}{2}{}{{x}}^{{2}}{+}{4}{}{{x}}^{{3}}{+}{10}{}{{x}}^{{4}}{+}{28}{}{{x}}^{{5}}{+}{84}{}{{x}}^{{6}}{+}{264}{}{{x}}^{{7}}{+}{858}{}{{x}}^{{8}}{+}{O}{}\left({{x}}^{{9}}\right)$ (2)
 > $\mathrm{seriestoalgeq}\left(s,y\left(x\right)\right)$
 $\left[{4}{}{x}{-}{2}{}{y}{}\left({x}\right){+}{{y}{}\left({x}\right)}^{{2}}{,}{\mathrm{ogf}}\right]$ (3)