Tetra-series

[Gottfried]

A curious result from study of the tetra-series. (text updated)

I considered the "reverse" of the tetra-series problem.

Instead of asking for the a_lternating s_um of powertowers of increasing p_ositive heights (asp)

Code:

asp(x,dxp) = dxp°0(x) - dxp°1(x) + dxp°2(x) - ... + ...

where dxp(x) = exp(x)-1 and dxp°h(x) is the h'th integer-iterate.

I asked for a function tf(x) where

Code:

asp(x,tf) = (e^x-1)/2 = tf°0(x) - tf°1(x) + tf°2(x) - ... +...

so I ask: can (e^x - 1)/2 be represented by a tetra-series of a function tf(x) and what would that function look like?



Using the matrix-operator-approach I got the result

Code:

tf(x) = x - x^2 + 2/3*x^3 - 3/4*x^4 + 11/15*x^5 - 59/72*x^6 + 379/420*x^7 - 331/320*x^8
       + 1805/1512*x^9 - 282379/201600*x^10 + 3307019/1995840*x^11 - 6152789/3110400*x^12
       + 616774003/259459200*x^13 - 3212381993/1117670400*x^14 + 54372093481/15567552000*x^15
       - 594671543783/139485265920*x^16 + 58070127447587/11115232128000*x^17
       - 1209735800444267/188305108992000*x^18 + 26776614379573099/3379030566912000*x^19
       - 209181772596680209/21341245685760000*x^20 + 1034961114326994557/85151570286182400*x^21
       - 80852235077445729119/5352384417988608000*x^22
       + 2210690796475549862239/117509166994931712000*x^23
       - 18624665294361841906483/793412278431252480000*x^24
       + 379264261780067802109819/12926008369442488320000*x^25
       - 6584114267874407529534167/179240649389602504704000*x^26
       + 5046681464320089079803469/109576837040799744000000*x^27
       - 326480035696597942691643978259/5646080455772478898176000000*x^28
       + 327920863401689931801359966641/4511103058030460180889600000*x^29
       - 418419411682443365665393881223739/4573325169175707907522560000000*x^30
       + 15798888070625404329026746075454779/137047310902965380295426048000000*x^31
       + O(x^32)

which can be determined to arbitrary many coefficients by a recursive process on rational numbers.

The float-representation is

Code:

tf(x) = 1.00000000000*x - 1.00000000000*x^2 + 0.666666666667*x^3 - 0.750000000000*x^4
      + 0.733333333333*x^5 - 0.819444444444*x^6 + 0.902380952381*x^7 - 1.03437500000*x^8
      + 1.19378306878*x^9 - 1.40068948413*x^10 + 1.65695596841*x^11 - 1.97813432356*x^12
      + 2.37715218038*x^13 - 2.87417649515*x^14 + 3.49265533084*x^15 - 4.26332874559*x^16
      + 5.22437379435*x^17 - 6.42433870711*x^18 + 7.92434807834*x^19 - 9.80176020073*x^20
      + 12.1543397362*x^21 - 15.1058348510*x^22 + 18.8129220299*x^23 - 23.4741329327*x^24
      + 29.3411740841*x^25 - 36.7333765543*x^26 + 46.0560972611*x^27 - 57.8241911808*x^28
      + 72.6919467774*x^29 - 91.4912883306*x^30 + 115.280540468*x^31 + O(x^32)

so I assume, that this series tf(x) has radius of convergence limited to about |x|<0.7

Moreover, the iterates of this function seem always to be of a similar form, so the alternating sum of the found coefficients of the iterated functions at like powers of x is divergent for each coefficient (but may be Euler-summed). So this result must be considered in more detail next, since I had inconsistency of the matrix-method with serial summation either for increasing positive or for increasing negative heights.

However, for x=1/2 or x=1/3 or smaller we can accelerate convergence of asp() by Euler-summation such that I get good (?) approximation to the six'th digit for x=1/3 using the truncated series with 31 terms only.

The process for the generation of these coefficients is a bit tedious yet; so I don't have -for instance - the function, whose iterations must be non-alternating summed to get the exp(x)-1 value (or (exp(x)-1)/2 or some other scalar multiple) which -as I guess- could have better range of convergence.

I'll post the result, if I got it.

Gottfried