compute the Borel transform of a generating function - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Discrete Mathematics : Combinatorics : gfun : gfun/borel

gfun[borel] - compute the Borel transform of a generating function

Calling Sequence

borel(expr, a(n), t)

Parameters

expr

-

linear recurrence with polynomial coefficients

a

-

name; recurrence name

n

-

name; index of the recurrence a

t

-

(optional) 'diffeq'; specify as a linear differential equation

Description

• 

The borel(expr, a(n)) command computes the Borel transform of a generating function.

• 

If an,n=0.. is the sequence of numbers defined by the recurrence expr, the borel function computes the recurrence for the numbers ann!.

• 

If an,n=0.. is the sequence of numbers defined by the recurrence expr, the procedure computes the recurrence for the numbers ann!.

• 

If t is specified as 'diffeq', expr is considered as a linear differential equation with polynomial coefficients for the function an. In this case, the function returns a linear differential equation satisfied by the Borel transform of an.

Examples

withgfun:

rec:=an=nan1+an2,a0=1,a1=1:

b:=borelrec,an

b:=an+n23n2an+1+n2+3n+2an+2,a0=1,a1=1

(1)

The invborel command is the inverse command.

invborelb,an

an+n2an+1+an+2,a0=1,a1=1

(2)

You can also perform Borel transforms on the corresponding differential equations.

deq:=rectodiffeqrec,an,fx:

newdeq:=boreldeq,fx,diffeq

newdeq:=fx2ⅆⅆxfx+1xⅆ2ⅆx2fx,f0=1,Df0=1

(3)

diffeqtorecnewdeq,fx,an

an+n23n2an+1+n2+3n+2an+2,a0=1,a1=1

(4)

See Also

gfun, gfun[diffeqtorec], gfun[invborel], rectodiffeq


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam