listtoratpoly - Maple Help

gfun

 listtoratpoly
 find a rational generating function
 seriestoratpoly
 find a rational approximation

 Calling Sequence listtoratpoly(l, x, [typelist]) seriestoratpoly(s, [typelist])

Parameters

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

Description

 • The listtoratpoly(l, x, [typelist]) command computes a rational function in x for the generating function 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.
 You should specify as many terms as possible in the list l.
 • The seriestoratpoly(s, x, [typelist]) command computes a rational function in x for the generating function 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.
 You should specify as many terms as possible in the series s.
 • 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 rational function. The second element is the generating function type for which a solution was found.
 • These functions are frontends to convert[ratpoly] which performs the actual computation.

Examples

If the input is the first few elements of the Fibonacci sequence, the function returns the generating series for the Fibonacci numbers.

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $l≔\left[1,1,2,3,5,8,13\right]$
 ${l}{≔}\left[{1}{,}{1}{,}{2}{,}{3}{,}{5}{,}{8}{,}{13}\right]$ (1)
 > $\mathrm{listtoratpoly}\left(l,x\right)$
 $\left[{-}\frac{{1}}{{{x}}^{{2}}{+}{x}{-}{1}}{,}{\mathrm{ogf}}\right]$ (2)
 > $\mathrm{seriestoratpoly}\left(\mathrm{series}\left(1+x+2{x}^{2}2!+3{x}^{3}3!+5{x}^{4}4!+8{x}^{5}5!+13{x}^{6}6!,x,8\right),\left['\mathrm{egf}'\right]\right)$
 $\left[{-}\frac{{1}}{{{x}}^{{2}}{+}{x}{-}{1}}{,}{\mathrm{egf}}\right]$ (3)