Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Rational sums of inverse powers of fixed points of e
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

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
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 Helms, Kassel

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

Possibly Related Threads...
Thread Author Replies Views Last Post
  tetration from alternative fixed point sheldonison 22 30,522 12/24/2019, 06:26 AM
Last Post: Daniel
  Thoughts on hyper-operations of rational but non-integer orders? VSO 2 774 09/09/2019, 10:38 PM
Last Post: tommy1729
  Inverse Iteration Xorter 3 2,759 02/05/2019, 09:58 AM
Last Post: MrFrety
  Inverse super-composition Xorter 11 13,776 05/26/2018, 12:00 AM
Last Post: Xorter
  Are tetrations fixed points analytic? JmsNxn 2 3,168 12/14/2016, 08:50 PM
Last Post: JmsNxn
  the inverse ackerman functions JmsNxn 3 6,148 09/18/2016, 11:02 AM
Last Post: Xorter
  Rational operators (a {t} b); a,b > e solved JmsNxn 30 41,432 09/02/2016, 02:11 AM
Last Post: tommy1729
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,637 03/19/2016, 10:44 AM
Last Post: fivexthethird
  Inverse power tower functions tommy1729 0 1,957 01/04/2016, 12:03 PM
Last Post: tommy1729
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 10,915 01/01/2016, 03:58 PM
Last Post: sheldonison

Users browsing this thread: 1 Guest(s)