Tetra-series

[Gottfried]

I've made a little progress with the problem of deviance of the serial summed alternaing series of powertowers of increasing heights ("Tetra-series"). Since T- and U-tetration can be mutually converted by shift of their parameter x, I can concentrate on the U-tetration here, which has the advantage, that its operator is a triangular matrix, whose integer-powers and eigensystem are easily and exact (within the unavoidable size-truncation for the actual computation) computable.

Code:

Denote the fixed base for

lb(x) = log(1+x)/log(b)
ub(x) = b^x - 1

The iterates

ltb(x,h) = ltb(lb(x),h-1)     ltb(x,0)= x
utb(x,h) = utb(ub(x),h-1)     utb(x,0)= x

The infinite alternating sums

ASLb(x) = x - ltb(x,1) + ltb(x,2) - ltb(x,3) + ... - ...
ASUb(x) = x - utb(x,1) + utb(x,2) - utb(x,3) + ... - ...

The matrix-approach suggests to compute the AS-series using the geometric-series of the U-tetration-matrices S1b and S2b, which I shall call MLb and MUb here. From here the conjecture was derived, that

Code:

ASLb(x) + ASUb(x) = x    // matrix-computation

since

Code:

MLb + MUb = I            // matrices

However, the computation of ASLb(x) along its partial sums and summation using Cesaro- or Euler-summation gives a different result:

Code:

ASLb(x) + ASUb(x) = x + db(x)  // serial-computation

The difference of the matrix- and serial-computation may then be expressed by db(x) only.

---------

In my previous posts I already mentioned, that the deviance between the two methods seem to be somehow periodic, wrt x as the variable parameter.

The first useful result was, that I found periodicity for db(x)

Code:

db(x) = - db(ltb(x,1)) = db(ltb(x,2)) = - db(ltb(x,3)) ...

for few bases and numerical accessible range for x. The plot showed also a sort of sinus-curve, but where the frequency was somehow distorted.

Today I could produce a very good approximation to a sinus-curve, using fractional tetrates for x.
I computed the k=0..32 fractional U-tetrates of x for height 1/16

Code:

x_k = ltb(1,k/16)

for base b=sqrt(2), and instead of x I used the index k as x-axis for the plot.
This provided a very good approximation to a sinus-curve for db(x_k)

In the plot I display the two near-lines for ASLb(x_k), ASUb(x_k) and db(x_k) and overlaid a sin-curve, whose parameters I set manually by inspection.
The curve for db(x_k) and sin() match very good; I added also a plot for their difference.

If I can manage to make my procedures more handy, I'll check the same for more parameters. The near match of the curves give apparently good hope that this line of investigation may be profitable...

Gottfried
[updated image]

   


Error-curve( deviation of db(x) from overlaid sinus-curve)