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+--- Thread: Mathematica program for tetration based on the series with q-binomial coefficients (/showthread.php?tid=1143)
Mathematica program for tetration based on the series with q-binomial coefficients - Vladimir Reshetnikov - 01/13/2017
A Mathematica program to compute tetration \( {^z a} \) for \( 1<a<e^{1/e} \) and \( \Re(z)>-2 \) based on the formula with q-binomial coefficients from http://mathoverflow.net/q/259278/9550.
Increase Terms and/or WorkingPrecision option values to get more digits. Examples of values computed using this program are http://goo.gl/MLfEMC and http://goo.gl/LVRkk7.
Code:
ClearAll[Tetration];
Options[Tetration] = {Terms -> 40, WorkingPrecision -> 100};
Tetration[_, -1, OptionsPattern[]] = 0;
Tetration[a_, n_Integer /; n >= 0, OptionsPattern[]] := Power @@ Table[a, {n}];
Tetration[a_ /; 1 < a < E^(1/E), z_ /; Re[z] > -2, OptionsPattern[]] :=
With[{q = -ProductLog[-Log[a]]},
With[{limit = SequenceLimit[N[Accumulate[Table[Sum[(-1)^(n + k) q^Binomial[n - k, 2] QBinomial[z, n, q] QBinomial[n, k, q] Tetration[a, k], {k, 0, n}], {n, 0, OptionValue[Terms]}]], OptionValue[WorkingPrecision]]]},
SetPrecision[limit, 3/4 Precision[limit]]]];
(* Examples:
Tetration[Log[3], 1/2]
Tetration[Log[3], 1/2, Terms -> 50, WorkingPrecision -> 150] *)