Re: 0.1010010001...

[Debra Woods]

I have having a terrible time finding info on this certain number.  It
is the number 0.1010010001... = Sum[1/(10^(n(n+1)/2)] I know it is
irrational. I can't seem to determine if it is algebraic or
transcendental.

I know that Sum[1/(10^(n!))] is transcendental because it is the
Liouville contstant and all Liouville numbers are transcendental.

I want to say that the first number is transcendental because when I
put it in Mathematica, it gives me
1/2 (-2 + 10^(1/8) EllipticTheta[2, 0, 1/Sqrt[10]])

and I want to say that Elliptic theta is probably transcendental.  
However, I also know that there are some that are algebraic.

I tried to show that the first number (does that thing have a name?)
is a Liouville number, but can't seem to do it perhaps because it
isn't and the 0's are not sparse enough?

If it is algebraic, then I would be interested in knowing the  
polynomial it comes from.

Someone told me it was algebraic, but that the second one wasn't, but
they couldn't prove it and it has been bothering me ever since.
Someone else claims that it is transcendental. ack.


Thanks!

[Jean-Paul Allouche-2]

Dear Debra

The number you are interested in is essentially the value of a theta
function at an algebraic argument, see e.g., the discussion bottom of
Page 490 and top of Page 491 (and references [5, 12, 30] there) of the
paper

   http://jtnb.cedram.org/cedram-bin/article/JTNB_2004__16_3_487_0.pdf

Look also at
   http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195510210

best
jean-paul