Science Talk

Gottfried Helms · Guest

A curious result from study of the tetra-series.

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)

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

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

so I ask for the function tf(x) whose tetraseries defines (e^x - 1)/2

Using the matrix-operator-approach I got the result

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

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 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
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

Guest

On 30 juin, 07:06, Gottfried Helms <he...@uni-kassel.de> wrote:

Gottfried Helms · Guest

Am 30.06.2008 17:00 schrieb alainverghote@gmail.com:

Gottfried Helms · Guest

Am 02.07.2008 20:00 schrieb Gottfried Helms:

Gottfried Helms · Guest

Am 03.07.2008 07:34 schrieb Gottfried Helms:

Gottfried Helms · Guest

Am 02.07.2008 20:00 schrieb Gottfried Helms:

amy666 · Guest

you might wanna shoot me for not starting a new thread, or not, but i post it here since i feel it is related.

recently i had an old idea again which relates to gottfrieds.

let T(x) be an arbitrary analytic function.
as the title says :

lim n-> oo (x - f(x) + f(f(x)) - f(f(f(x))) + ... - ... ) / n = T(x)

where the sum is taken n times of course , thus an anternating average.

since /n implies that else it doesnt converge , that implies the iterations of f do not converge.

if the iterations are bounded then this can be related to a periodic (analytic) function g where the n th iterate of f = g(n).

if g also has an addition formula , we might even get a closed form for f(x).

and g would then probably be ( here we go again :p )

an inverse hypergeo function !

i believe that with a very good understanding of ergodic theory, umbral calculus and the concepts of the (periodic) inverse hypergeo functions we can solve the equations of type

lim n-> oo (x - f(x) + f(f(x)) - f(f(f(x))) + ... - ... ) / n = T(x)

note :

the limit might perhaps in simple cases be equal to the integral over its period ( contour method ! ) of g.

( this makes sence , if it sounds strange stop for a moment and think about it ! )

it is striking how i once again end up with inverse hypergeo and contour integration !!

however what might interest gottfried more is if this also related to his OP.

we can of course rewrite f(f(x)) as g(2) etc but will that help ?

f(x) = g(y)

f(f(x)) = g(y+1)

x - g(y) + g(y+1) - g(y+2) + ... = T(x)

my guess is the following :

the + and - together with the fact that lim g(oo) is probably = oo leads me to the consideration that the left side is an analytic continuation of a similar series ( of g ).

then also T(x) must be an analytic continuation of something or related to a self-similar looking function ( if it has no poles e.g. ).

( see my concept sum over the reals of f(x) , which is somewhat related )

both approaches are hard but very different.

do the 2 methods relate ?

does the second also relate to contour integration and the concept of inverse hypergeometric functions ??

i admit i dont know the answer.

***

regards

tommy1729

amy666 · Guest