Today I experienced one of these "click"s ...: "ahhh - why didn't I see this before...?"

In the dxp-Bell/Carlemann-matrix the coefficients behave badly when the base-parameter approaches the limits of the range of convergence; for the iterated exp-function if b-> eta and for the iterated dxp-function if base t->e.

Today I found an extremely useful transformation which allows to look at the powerseries for bases arbitrarily near at this limit.

The problem for the construction of coefficients (for dxp) is based on that we have denominators containing the expression (u-1) and its powers where u is the log of the base t, so that for u near 1 (thus base t near exp(1) or for exp-tetration b near eta) the numerical evaluation of the expressions for the coefficients diverges badly.

Now I realized that it is something trivial to remove that (u-1)-values from the coefficients and use a transformed x-value instead: dxp_t°h (x) becomes then dxpnew_t°h(x') instead where x'=x/(u-1) .

This may not seem to be a real progress, because even if the coefficients of the powerseries for dxpnew behave better, then the powers of its parameter x' behave reciprocally bad and simply compensate the new kindness of the dxpnew-coefficients.

But in fact we have even two advantages:

- if x/(u-1) gives a value outside of a reasonable range of convergence then by repeated integer-height-iteration we can reduce x/(u-1) to x' until x' is small enough and we can use the number of required integer-iterations as an additive height-constant. So we can easily compute the schroeder-functions for any base arbitrarily near the limit case

[Update] Hmm, second thought: this is nothing new by this "(u-1)-replacement". That normalizing of the "x" using the integer-height-tetration must/should be done also in the original regular tetration and its schroeder-function. I don't see actually how the "(u-1)-replacement" adds something special here... I'll comment on this later again [/update]

- the coefficients of the eigenmatrices (resp. for its schroeder-function) for dxpnew() are nicely decreasing. That propagates also to the carlemann-matrices for the fractional iterates - at least as far as I can look at truncation to 64 terms (which is my current upper limit for the symbolic representation of these matrices). So maybe we can state that we're going to overcome the notorious zero-radius of convergence for fractional iterates by this simple transformation.

[update] Also this is questionable in a second view. The "(u-1) replacement" is an extraction of factors from the coefficients of the regular tetration formal powerseries for a fractional height. But the growthrate of that coefficients is at least hypergeometric and the factor-extraction removes just the consecutive powers of (u-1). So the hypergeometric sequence is reduced by a geometric sequence, and this does not change its basic character. Only the index where the growth of coefficients begins is shifted to a later position. But than be checked if I use bigger matrices/more terms of the powerseries for fractional heights [/update]

And finally: I took log(t) = u = 1.0 -1e-99, that means t=exp(1) - eps with eps->zero. It is extremely astonishing that the powerseries of the schroeder-function approximates 1x-1/2x^2+1/4x^3-...+... = x/(1+x/2) and of the inverse schroeder-function just 1x+1/2x^2+1/4x^3+...+... = x/(1-x/2)

That seems to allow to formulate a new(?) limit-theorem...

(I've not yet elaborated on the latter, possibly I'll post something later today.)

Gottfried

In the dxp-Bell/Carlemann-matrix the coefficients behave badly when the base-parameter approaches the limits of the range of convergence; for the iterated exp-function if b-> eta and for the iterated dxp-function if base t->e.

Today I found an extremely useful transformation which allows to look at the powerseries for bases arbitrarily near at this limit.

The problem for the construction of coefficients (for dxp) is based on that we have denominators containing the expression (u-1) and its powers where u is the log of the base t, so that for u near 1 (thus base t near exp(1) or for exp-tetration b near eta) the numerical evaluation of the expressions for the coefficients diverges badly.

Now I realized that it is something trivial to remove that (u-1)-values from the coefficients and use a transformed x-value instead: dxp_t°h (x) becomes then dxpnew_t°h(x') instead where x'=x/(u-1) .

This may not seem to be a real progress, because even if the coefficients of the powerseries for dxpnew behave better, then the powers of its parameter x' behave reciprocally bad and simply compensate the new kindness of the dxpnew-coefficients.

But in fact we have even two advantages:

- if x/(u-1) gives a value outside of a reasonable range of convergence then by repeated integer-height-iteration we can reduce x/(u-1) to x' until x' is small enough and we can use the number of required integer-iterations as an additive height-constant. So we can easily compute the schroeder-functions for any base arbitrarily near the limit case

[Update] Hmm, second thought: this is nothing new by this "(u-1)-replacement". That normalizing of the "x" using the integer-height-tetration must/should be done also in the original regular tetration and its schroeder-function. I don't see actually how the "(u-1)-replacement" adds something special here... I'll comment on this later again [/update]

- the coefficients of the eigenmatrices (resp. for its schroeder-function) for dxpnew() are nicely decreasing. That propagates also to the carlemann-matrices for the fractional iterates - at least as far as I can look at truncation to 64 terms (which is my current upper limit for the symbolic representation of these matrices). So maybe we can state that we're going to overcome the notorious zero-radius of convergence for fractional iterates by this simple transformation.

[update] Also this is questionable in a second view. The "(u-1) replacement" is an extraction of factors from the coefficients of the regular tetration formal powerseries for a fractional height. But the growthrate of that coefficients is at least hypergeometric and the factor-extraction removes just the consecutive powers of (u-1). So the hypergeometric sequence is reduced by a geometric sequence, and this does not change its basic character. Only the index where the growth of coefficients begins is shifted to a later position. But than be checked if I use bigger matrices/more terms of the powerseries for fractional heights [/update]

And finally: I took log(t) = u = 1.0 -1e-99, that means t=exp(1) - eps with eps->zero. It is extremely astonishing that the powerseries of the schroeder-function approximates 1x-1/2x^2+1/4x^3-...+... = x/(1+x/2) and of the inverse schroeder-function just 1x+1/2x^2+1/4x^3+...+... = x/(1-x/2)

That seems to allow to formulate a new(?) limit-theorem...

(I've not yet elaborated on the latter, possibly I'll post something later today.)

Gottfried

Gottfried Helms, Kassel