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gfun[listtodiffeq] - find a linear differential equation for the generating function

gfun[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

withgfun:

l:=1,2,6,22,91,408,1938,9614,49335,260130,1402440,7702632,42975796,243035536,1390594458,8038677054,46892282815,275750636070,1633292229030,9737153323590:

listtodiffeql,yx

27x4x2ⅆ2ⅆx2yx+108x2+18xⅆⅆxyx12+60x+12yx,y0=1,Dy0=2,ogf

(1)

s:=seriesⅇx1x,x,7

s:=1+32x+118x2+5348x3+115128x4+29473840x5+3141146080x6+Ox7

(2)

seriestodiffeqs,yx

2x+2ⅆⅆxyx+2x3yx,y0=1,ogf

(3)

See Also

gfun, gfun/gftypes, gfun/parameters, series


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