+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: Tetration: Progress in fractional iteration? (/showthread.php?tid=197)
Tetration: Progress in fractional iteration? - Gottfried - 08/13/2008
Hi -
recently I posted this in sci.math.research. I think a copy of this here would be good, too (I've posted a similar msg already in the thread "matrix-method", but may be not everyone expects such general things there) The progress is mentioned in part 3.
------------------------------------------------------------
Tetration: Progress in fractional iteration?
In 1958 I.N.Baker proved in [1], that the powerseries for
fractional iterates of the function exp(x)-1 have
convergence-radius zero. P.Erdös / E.Jabotinsky followed
in [2] with the stronger statement "The function exp(x) - 1
was shown by I. N. Baker [L] to have no real non-integer iterates."
Attempts to define fractional iterates of exp(x)-1 or more
general t^x-1 in the context of the "tetration"-discussion
are since rated with a portion of suspicion...
However - even if a series has convergence-radius zero it
may be summed using a technique of divergent summation; one
other example for zero-convergence-radius is the series
f(x) = 0! - 1!x + 2!x^2 - 3!x^3 + ... - ...
to which a summation-method was applied by L.Euler.
The extremely simple Euler-transformation, for instance,
allows to sum the alternating geometric series up to any
order by transforming the original series of coefficients
a_k into coefficients b_k, which form then a conventionally
summable series.
I seem to have found a similar simple procedure for the
functions U(x) = exp(x)-1 and Ut(x) = t^x - 1, and especially
their fractional iterates, just using the Stirlingnumbers 2nd kind
analoguously to Euler's binomial-coefficients.
This transformation seems to transform the diverging sequences
of coefficients a_k (having also nonperiodic change of sign) even
of fractional iterates into the converging sequence of b_k,
if the base t is greater than exp(1.5) - which are the especially
difficult cases since the iterates diverge for bases >2.
A short technical report is at
http://go.helms-net.de/math/tetdocs/CoefficientsUtFractionalHeight.htm
It reflects only some initial findings, but I think, it
gives already a wider perspective - let's see.
Comments/critics/corrections are much appreciated -
Gottfried Helms
[1] Baker, Irvine Noel; Zusammensetzungen ganzer Funktionen
1958; Mathematische Zeitschrift, Vol 69, Pg 121-163,
[2] Erdös, Paul, Jabotinsky, Eri; On analytical iteration
1961; J. Anal. Math. 8, 361-376
====================================================