Tetration: Progress in fractional iteration? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/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 ====================================================