generatingfunctions

[wouter meeussen]

dear Mitch,

my apologies are in order:
you originally asked for
z+z^2+z^4+z^8+.. z^[2^n]
and I misread as
z+z^4+z^9+z^16+ ..z^[n^2]

which actually is
1/2*(1 + EllipticTheta[3, 0, z]).

i have no relevant input on the original question.

Wouter.

----- Original Message -----
From: "Mitch Harris" <maharri at cs.uiuc.edu>
To: "Meeussen Wouter (bkarnd)" <wouter.meeussen at vandemoortele.com>
Cc: <ralf at ark.in-berlin.de>; <seqfan at ext.jussieu.fr>
Sent: Tuesday, April 29, 2003 5:10 AM
Subject: RE: generatingfunctions


> On Mon, 28 Apr 2003, Meeussen Wouter (bkarnd) wrote:
> >Ralf Stephan [mailto:ralf at ark.in-berlin.de] wrote:
> >>Mitch Harris:
> >>> >Possibly someone knows of references to special cases?
> >>>
> >>> I think this is hard in general: take r(z) = 1/(1-2z), so r(n) = 2^n.
I
> >>> don't know of a 'good' (rational function or even special function)
> >>> representation of the ogf
> >>>
> >>>   \sum z^n [n=2^k]
> >>
> >>[log2(n)]-[log2(n-1)], n>1, would be cheating?
>
> That I didn't think of.
> but what is the gf of floor(log n)? there's the recurrence, but no closed
> form for it (that I know of).
>
> >In[8]:=
> >Series[EllipticTheta[4, 0, x], {x, 0, 37}]
> >
> >Out[9]=
> >1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36
> >
> >comes pretty close, nah?
>
> touche! It would be lame of me to say that I considered that immediately
> after I posted, so I won't.
>
> However, what about higher powers, z^(n^3), .., can you get those with
> similar functions? What about combinations z^(2^n + 3^n + n^17) ?
>
> --
> Mitch Harris
>
> Department of Computer Science
> University of Illinois at Urbana-Champaign
> http://www.uiuc.edu/~maharri
>
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