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gfun

  

borel

  

compute the Borel transform of a generating function

 

Calling Sequence

Parameters

Description

Examples

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:

recan=nan1+an2,a0=1,a1=1:

bborelrec,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.

deqrectodiffeqrec,an,fx:

newdeqboreldeq,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

 


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