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Rational sums of inverse powers of fixed points of e
#13
jaydfox Wrote:Anyway, in the meantime, I wanted to try to figure out how to calculate the numerators. It seems to me that they deserve their own Sloane sequence, considering how surprising it was (to me) to get rational sums.

Hmm, this may not be of much help, at least I didn't find out a useful simplification. Anyway.

If I use my formula for complex fixpoint for real base s, with the single independent parameter beta:

u = alpha + beta*i
where
alpha=beta/sin(beta)*cos(beta)

t = exp(u) = a + b*i
s = exp(u/t)

and in your example s=e, then these formulae can possibly be reversed to determine the allowed values for beta.

The only thing I know already is, that the different allowed beta_k are roughly periodic at k*2*pi+eps_k with eps_k decreasing towards zero.

So for any t of all t_k, omitting the index,
t = exp( alpha + beta*i)
= exp(alpha)*cos(beta) + exp(alpha)*sin(beta)*I
and
1/t = exp(-alpha - beta*i) = exp(-alpha)/(cos(beta)+sin(beta)*i)
= exp(-alpha)*(cos(beta)-sin(beta)*i)

Now you consider the sum of 1/t and 1/conj(t) as one term v:
v=1/t + conj(1/t) = 2*exp(-alpha)*cos(beta)
v=2 * exp(-beta/sin(beta)*cos(beta))*cos(beta)
and ask, whether the sum of all v_k add up to a rational...

I don't know, how to proceed from here; the most difficult thing is surely the reverse determination of the possible beta's from the given base-parameter s=e.

Hmmm ... an apple without vitamins...

Gottfried
Gottfried Helms, Kassel
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Messages In This Thread
RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/22/2007, 06:25 PM

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