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gfun

  

invborel

  

compute the inverse Borel transform of a generating function

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

invborel(expr, a(n), t)

Parameters

expr

-

linear recurrence with polynomial coefficients

a

-

name; recurrence name

n

-

name; index of the recurrence a

t

-

(optional) name; 'diffeq'

Description

• 

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

  

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

• 

If t is specified as 'diffeq', expr is considered a linear differential equation with polynomial coefficients.  The function returns a linear differential equation satisfied by its inverse Borel transform.

• 

Laplace can be used as a synonym of invborel, that is, the Laplace(expr, a(n)) command can be used instead.

Examples

withgfun:

recan=nan1+an2,a0=0,a1=1:

deqinvborelrec,an

deq:=n23n2an+n24n4an+1+an+2,a0=0,a1=1

(1)

rec2rectodiffeqdeq,an,ft

rec2:=2t2t+1ft+4t33t2ⅆⅆtft+t4t3ⅆ2ⅆt2ftt

(2)

deq2borelrec2,ft,'diffeq'

deq2:=t2t+1ftⅆⅆtftt2t

(3)

diffeqtorecdeq2,ft,an

an+n2an+1+an+2,a0=0,a1=1

(4)

See Also

gfun

gfun[borel]

gfun[Laplace]

gfun[rectodiffeq]

 


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