generatingfunctions

[Karol PENSON]

 Dear Martin, It seems that the function F(z) as defined in your mail
 is the solution of the following functional equation:

  F(z)=z+F(z^2), with F(0)=0.

 Can it help to understand what is going on ? 

  Karol


On Tue, 29 Apr 2003, Martin Klazar wrote:

> Date: Tue, 29 Apr 2003 14:49:24 +0200 (CEST)
> From: Martin Klazar <klazar at kam.mff.cuni.cz>
> To: seqfan at ext.jussieu.fr
> Subject: Re: generatingfunctions
> 
> 
> As for the possible representations of sparse ("lacunary")
> power series, such as
> 
> F(z)=z+z^2+z^4+z^8+...,
> 
> the following theorem of Lipshitz and Rubel (1986) applies
> and shows that F(z) satisfies no algebraic differential
> equation over C(x). (Theta functions, mentioned in
> this discussion, do satisfy alg. dif. equations).
> 
> Thm.
> If a_0.x^{n_0}+a_1.x^{n_1}+..., where n_0<n_1<... and
> all a_i are nonzero, is dif. algebraic
> then lim inf n_{i+1}/n_i=1.
> 
> (Their actual result is somewhat stronger.)
> 
> 
> Martin Klazar
> 

-- 
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Karol A. PENSON
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