02/18/2008, 07:19 PM
I started rethinking the inconsistency about the tetraseries with increasing negative heights.
First - I don't remember whether I've already linked my compilation concerning this problem. Here is the link tetraseries-problem
Then I realized, that using a fixpoint we'll have a special case.
Recall the definition
ALU (b,x) = x - log(1+x)/log(b) + log(1+log(1+x)/log(b))/log(b) - ... + ...
which means the infinite alternating sum of towers of negative heights.
If x is a fixpoint of b, where b^t-1=t or b=(1+t)^(1/t), then clearly this simplifies to
ALU (b,t) = t - t + t - ... + ... = t * eta(0) = t/2 //by Euler-or Cesaro-summation. and the conjecture
ASU(b,t) + ALU(b,t) - t =0
holds for this case.
Don't know how to make this helpful for reparature of my conjecture, but perhaps it gives an idea.
Gottfried
@Andrew (I just also reread your note in the other thread): the needed matrix is simple to compute. Just construct the triangular Bell-matrix U (for decremented iterated exponentiation for a base b), and compute
MU = (I + U)^-1
and
ML = (I + U^-1)^-1
First - I don't remember whether I've already linked my compilation concerning this problem. Here is the link tetraseries-problem
Then I realized, that using a fixpoint we'll have a special case.
Recall the definition
ALU (b,x) = x - log(1+x)/log(b) + log(1+log(1+x)/log(b))/log(b) - ... + ...
which means the infinite alternating sum of towers of negative heights.
If x is a fixpoint of b, where b^t-1=t or b=(1+t)^(1/t), then clearly this simplifies to
ALU (b,t) = t - t + t - ... + ... = t * eta(0) = t/2 //by Euler-or Cesaro-summation. and the conjecture
ASU(b,t) + ALU(b,t) - t =0
holds for this case.
Don't know how to make this helpful for reparature of my conjecture, but perhaps it gives an idea.
Gottfried
@Andrew (I just also reread your note in the other thread): the needed matrix is simple to compute. Just construct the triangular Bell-matrix U (for decremented iterated exponentiation for a base b), and compute
MU = (I + U)^-1
and
ML = (I + U^-1)^-1
Gottfried Helms, Kassel
