Hello Dmitrii -

I still wish to learn/understand your specific method. This time I followed the link to your wiki http://mizugadro.mydns.jp/t/index.php/Tetration and looked at the entry for the tetration to base b=exp(1/e). I do not understand how you arrive at the polynomials, and especially how at the set of power-series in the table with the increasing m.

However, I could find the coefficients of the first row (m=0) by my standard procedure, and my method allows 12,16 or 20 correct digits for them. So if I knew, how our methods are related, and if they come out to be identical (at least for that exact base), then you could take something useful from my method, at least for the more accurate computation of the m=0 coefficients.

The way how I arrive at that many digits is some well experienced method of Noerlund-summation of alternating divergent series. Unfortunately not many people are used to such methods and thus are unable to apply them if their projected coefficients stem from formal expressions which represent divergent series. Or from functions, whose formal expressions as powerseries have zero-range of convergence, but have taylor-coefficients with alternating signs - in the latter case they can in many nontrivial cases be summed for an actual given argument x or z outside the range of convergence.

What I'm doing, in short, is using the concept of Carlemanmatrices.

Carlemanmatrices are invented for the formal manipulation of composition (and thus also iteration) of functions which have a power series representation (=are analytic). For the given function f(x), the exponential with base exp(1/e) and its conjugation to some g(x) which is defined to satisfy f(x)=g(x/L-1)+1)*L (with L its fixpoint) the Carleman-matrix G is of special simple form (triangular with units in the diagonal) which allows easily computation of its (Matrix-)logarithms and thus arbitrary fractional powers by G^h = Exp(h * Log(G)) and where G^h contains then the coefficients of the power-series for the h'th iterate (g°h(x/L-1)+1)*L even for fractional or complex iteration-height h.

Tetration in your terminology is then the evaluation at x=1 where the iterationheight h is given as the p+qi - argument in your description.

If I leave -in my procedures - the iteration-height parameter h indetermined in the resulting two-variable power series, but evaluate for the case x=1 I arrive at the coefficients c very similar to that in your article, but can supply many more digits precision - however only because of the technique of Noerlund-summation for the series where the x=1 - argument is involved/evaluated.

If this is interesting for you then ask for more information (I can also supply the Pari/GP-toolbox for that computations).

See here for instance the approximations of the first few coefficients c (after my carlemanmatrix/Noerlundsummation - concept). With powerseries of 64 terms I get in the rows 59 to 64 the partial sums of the summations for the c-coefficients, when that summations are understood as summing columns of data.

Here are the values:

Where your coefficients were given as

The data at m=0 match suspiciously well, however I've no further idea yet how to reproduce the rows at m=1,m=2 and so forth.

Also, to exclude the possibility of a pure incidence/of a "false positive", it would perhaps be good if I could see more of the coefficients in that first row of your table...

I still wish to learn/understand your specific method. This time I followed the link to your wiki http://mizugadro.mydns.jp/t/index.php/Tetration and looked at the entry for the tetration to base b=exp(1/e). I do not understand how you arrive at the polynomials, and especially how at the set of power-series in the table with the increasing m.

However, I could find the coefficients of the first row (m=0) by my standard procedure, and my method allows 12,16 or 20 correct digits for them. So if I knew, how our methods are related, and if they come out to be identical (at least for that exact base), then you could take something useful from my method, at least for the more accurate computation of the m=0 coefficients.

The way how I arrive at that many digits is some well experienced method of Noerlund-summation of alternating divergent series. Unfortunately not many people are used to such methods and thus are unable to apply them if their projected coefficients stem from formal expressions which represent divergent series. Or from functions, whose formal expressions as powerseries have zero-range of convergence, but have taylor-coefficients with alternating signs - in the latter case they can in many nontrivial cases be summed for an actual given argument x or z outside the range of convergence.

What I'm doing, in short, is using the concept of Carlemanmatrices.

Carlemanmatrices are invented for the formal manipulation of composition (and thus also iteration) of functions which have a power series representation (=are analytic). For the given function f(x), the exponential with base exp(1/e) and its conjugation to some g(x) which is defined to satisfy f(x)=g(x/L-1)+1)*L (with L its fixpoint) the Carleman-matrix G is of special simple form (triangular with units in the diagonal) which allows easily computation of its (Matrix-)logarithms and thus arbitrary fractional powers by G^h = Exp(h * Log(G)) and where G^h contains then the coefficients of the power-series for the h'th iterate (g°h(x/L-1)+1)*L even for fractional or complex iteration-height h.

Tetration in your terminology is then the evaluation at x=1 where the iterationheight h is given as the p+qi - argument in your description.

If I leave -in my procedures - the iteration-height parameter h indetermined in the resulting two-variable power series, but evaluate for the case x=1 I arrive at the coefficients c very similar to that in your article, but can supply many more digits precision - however only because of the technique of Noerlund-summation for the series where the x=1 - argument is involved/evaluated.

If this is interesting for you then ask for more information (I can also supply the Pari/GP-toolbox for that computations).

See here for instance the approximations of the first few coefficients c (after my carlemanmatrix/Noerlundsummation - concept). With powerseries of 64 terms I get in the rows 59 to 64 the partial sums of the summations for the c-coefficients, when that summations are understood as summing columns of data.

Here are the values:

Code:

`...: ...`

59: 1.000000000000000 0.6110954537716500 -0.2317026144766614 0.09178128765991140 -0.03756492168250494 0.01577372205052486

60: 1.000000000000000 0.6110954537716508 -0.2317026144767069 0.09178128766078397 -0.03756492169126225 0.01577372210673970

61: 1.000000000000000 0.6110954537716513 -0.2317026144767332 0.09178128766130164 -0.03756492169661943 0.01577372214214125

62: 1.000000000000000 0.6110954537716514 -0.2317026144767481 0.09178128766160907 -0.03756492169989380 0.01577372216440198

63: 1.000000000000000 0.6110954537716515 -0.2317026144767563 0.09178128766179094 -0.03756492170189427 0.01577372217838127

64: 1.000000000000000 0.6110954537716517 -0.2317026144767610 0.09178128766189732 -0.03756492170311400 0.01577372218715045

...: ...

Where your coefficients were given as

Code:

`m c_{m,0} c_{m,1} c_{m,2} c_{m,3} c_{m,4}`

0 1 0.61061 -0.23171 0.09225 -0.03757

1 0 0.69521 0.41315 -0.16027 0.07007

2 0 -0.57851 0.18323 0.49162 -0.15216

3 0 0.64730 -0.62933 -0.51128 0.51372

4 0 -0.84098 1.23261 0.42470 -0.97551

5 0 1.19090 -2.12653 -0.06895 1.57684

The data at m=0 match suspiciously well, however I've no further idea yet how to reproduce the rows at m=1,m=2 and so forth.

Also, to exclude the possibility of a pure incidence/of a "false positive", it would perhaps be good if I could see more of the coefficients in that first row of your table...

Gottfried Helms, Kassel