Henryk -
I crosschecked some results of your method against the diagonalization-method.
I got some differences, which look somehow suspicious to me.
Using, for instance,
u = 0.9 (to be near at a critical value)
t = exp(u) = 2.45960311116
b = t^(1/t) = 1.44182935647
I got
y = b°0.5(1) = 1.25590667...
where in these digits no difference appear.
But I got differences in less significant digits, which -as I said- look suspicious: either my summation-methods are not perfect (in fact they give different approximations - I'm still searching for better ones), or I need simply more terms (opposite to the impression).
They stabilize (using 96 terms) with differences in the last partial sums of order 1e-18 to 1e-22. So they seem to give usable results.
However, the difference to your binomial-method
y<binomial> - y<matrix> ~ 0.00000000259698761292
is so large compared to the summation-internal inaccurcy, that this seems to be systematic. Different summation-methods gave different result even worse - so in any case I've to improve my summation for these type of divergent series. For t=2, b=sqrt(2) it was much better, but possibly things were simply not visible...
I hope I can locate (and possibly remove) the source of the difference...
Gottfried
I crosschecked some results of your method against the diagonalization-method.
I got some differences, which look somehow suspicious to me.
Using, for instance,
u = 0.9 (to be near at a critical value)
t = exp(u) = 2.45960311116
b = t^(1/t) = 1.44182935647
I got
y = b°0.5(1) = 1.25590667...
where in these digits no difference appear.
But I got differences in less significant digits, which -as I said- look suspicious: either my summation-methods are not perfect (in fact they give different approximations - I'm still searching for better ones), or I need simply more terms (opposite to the impression).
They stabilize (using 96 terms) with differences in the last partial sums of order 1e-18 to 1e-22. So they seem to give usable results.
However, the difference to your binomial-method
y<binomial> - y<matrix> ~ 0.00000000259698761292
is so large compared to the summation-internal inaccurcy, that this seems to be systematic. Different summation-methods gave different result even worse - so in any case I've to improve my summation for these type of divergent series. For t=2, b=sqrt(2) it was much better, but possibly things were simply not visible...
I hope I can locate (and possibly remove) the source of the difference...
Gottfried
Gottfried Helms, Kassel