Gottfried Helms  Univ Kassel 04  2008 

Powerseries for U_{t} at fractional heights
Coefficients of powerseries for fractional iteration U_{t}^{oh}(x)
Abstract:
Iteration of the function U_{t}(x) = t^{x}–1 to fractional heights provides powerseries, whose coefficients mostly strongly diverge and oscillate. I.N. Baker proved in [Bak], that fractional iterates of the function exp(x) – 1 have convergenceradius zero and thus cannot conventionally be summed. The divergence and irregular oscillating of signs is so nasty, that even common techniques for divergent summation like Cesaro and Eulersum give no satisfying result  if their results are usable at all.
Here I give a graphical overview in three examples of different characteristics, for bases t=2, t=3, and t=exp(2) .
[Update:]
It seems, that a new seriestransformation using the Stirlingnumbers 2^{nd} kind is suitable to convert the powerseries with diverging coefficients into such of convergent coefficients – thus an expanding of the range of convergence for x from zero to finite values is now realized (hopefully to infinity, must be proven). See chap 3.
Gottfried Helms, DKassel Ver 2.2; 6.02.09
Contents
1.1. Definition of the basefunction U_{t}(x) and U_{t}^{oh}(x)
3. Achieving convergence by Stirlingtransformation
3.2.2. Using second power, U^{2}
3.2.3. Using third power, U^{3}
3.4. Provisorial conclusion / Impulse to proceed
Define the iterable function
U_{t}(x)
=t^{x} – 1
U_{t}°^{1}(x)
= U_{t}(x)
U_{t}°^{0}(x)
= x
U_{t}°^{h}(x)
= U_{t}°^{h1}(U_{t}(x))
then the functions with different h have different powerseriesexpansion. I'll denote the coefficients of a powerseries for a certain height h as a_{k}, k beginning at zero, such that
U_{t}^{°h}(x) = a_{1} x + a_{2} x^{2} + ... // there is no constant term a_{0}
For fractional h we have alternating divergent sequences of coefficients a_{k}. with a morethangeometric growth. Thus to that powerseries we cannot conventionally assign a value; however the theory of divergent summation shows, that an assignment of a value can be meaningful (and consistent) for some classes of series. Thus a suitable method for summation of divergent series must be applied.
Unfortunately, the rate of growth of the coefficients is hypergeometric. That means, that the powerseries does not converge for whatever small absolute value of x. For instance, if f(x) has factorials as coefficients
f(x) = 1! x – 2! x^{2} + 3! x^{3} – 4! x^{4} + …
then whatever small value x>0 is considered, the series eventually diverges, and the radius of convergence (wrt x) is said to be zero.
For series, whose coefficients do not grow so fast, for instance all ratios of consecutive coefficients are a constant value q (thus themselves form a geometric sequence) we can assign a value using Euler or Borelsummation of appropriate order.
But the coefficients of the powerseries for fractional heights increase even faster than the factorials in the example above, so Eulersummation, for instance, is not applicable (i.e. cannot supply arbitrarily precise approximation) and another general method of summation must be found. I've usually applied a method based on the idea of Rieszsummation (see a general description for instance in [Vol]) which gives often good approximated meaningful values – but in general without convincing power yet.
Update: In chap 3 I propose a summation method, which seems to solve this problem. (see also Appendix 4.1 for the matrixdefinitions required later)
To understand the characteristic of the growth of the coefficients I produced some plots to compare them visually for different heights.
The coefficients a_{k} for integer heights h=1,2,... form convergent sequences; for instance for h=1 we have the simple exponentialseries (writing u for log(t))
U_{t}°^{1}(x) =
a_{1} x + a_{2} x^{2} + a_{3} x^{3} + …
= ux +
u^{2}/2! * x^{2} + u^{3} / 3! *x^{3} + …
And the sequence of coefficients
(a_{1},a_{2},a_{3},a_{4},…) = (u, u^{2}/2!, u^{3}/3! ,… )
eventually tends to zero, thus the radius of convergence of U_{t}°^{1}(x) wrt x is even infinite.
But for noninteger heights, for instance h=0.975, 1.65, 1.875 the sequences of coefficients eventually diverge. To get an idea about the characteristics of that convergent/divergent behave of the coefficients I show trajectories of a_{k}, k=1..128 for some example heights h (integer and fractional) and also for three different bases t.
In [Helms0712] I show the coefficients for fractional iterates of the function U(x)=exp(x)1, thus t=exp(1), u=1. The characteristics may be slightly different from the herediscussed series, since with t=exp(1) I used the matrixlogarithm for fractional powers and for other t I use diagonalization. The general overview over the tables of coefficients suggest, that the trajectory of absolute values of a_{k} starts descending to a global minimum at a certain index k, from where it continuously increases with morethangeometric rate. The interesting impression is, that the index k disappears linearly with the logarithm of d (in h=1 + d, where d<1/2) to infinity.
Example table
d = 1e4 k_min=19
… k_min=19
d = 1e10 k_min=19
d = 1e15 k_min=23
d = 1e20 k_min=25
d = 1e25 k_min=29
...
...
The move seems indeed to be linear in k with log(d), thus the infinite radius of convergence of the h=1 iterate is due to the fact, that d>10^{oo} and the index k of the global minimum accordingly moves away to the infinity.
Here is a zoomed image for the coefficients of powerseries for d=10^{11}, 10^{12},10^{13} ,… 10^{25} containing the ak in columns. The roughly smallest absolute values a_{k} (the global minima for each height) are marked by the red line
Coefficients of powerseries for different heights. The first 64 terms of each powerseries are displayed in one column
1+10^{11} 1+10^{15} 1+10^{20} 1+10^{24} 
= height 


In the following plot I display the coefficents for the powerseries using t=exp(2)
I computed 128 coefficients for all example heights h.
We see the convergence of the sequence of coefficients for integer h whose graphs soon converge to the xaxis, while the a_{k}sequences for fractional h have a smooth graph at the beginning but are eventually oscillating in sign and are also diverging in absolute values. Note that the yscale is exponential.
Graph 1


In the above type of plot we cannot compare more than some example heights. To have the growthcharacteristics in a greater and more continuous overview, I show another type of plot. Here I use 161 powerseries for heights in the interval h=0..2 in equal steps.
However, this mass of coefficients is difficult to print and to evaluate, if given in numbers. So here is a bitmap, whose vertical direction indicates the coefficients a_{k} of one powerseries (for one specific height parameter h) and the subsequent heights are the subsequent columns of one pixel for each height h. Thus the vertical column of one pixel at left border represent the coefficients of the powerseries for height h=0, the vertical line in the middle of the graph represents the coefficients of the powerseries for h=1 and the right border represents the powerseries for height h=2.
Positive coefficients are marked red and negative coefficients are marked green, with brightnessvalue by b_{k} = log(abs(a_{k}))/log(10), with clipping at b_{k}=min(max(b_{k},16),16)+sign(a_{k})*16; so the brightness of the color red/green resembles roughly the index of the most significant digit of the interval 1e16<abs(a_{k})<1e16 (brightness 0<=b_{k}<=32)
Plot 1: Coefficients A of powerseries of different heights h 0<=h<=2


In the middle there is the vertical line for the convergingtozero coefficients of the powerseries at h=1, and at the right border a vertical line at h=2.
In the darker area the absolute values of coefficients are smaller, in the lighter areas greater. We see an increasing of light to the bottom which indicates growth of the coefficients.
There is some more finestructure visible: the antidiagonal lines are very obvious, but there are some brightnessareas with a very fine topology (difficult to discern in this bitmap)
"Pixelizing" gives a better view into the general structure:
Plot 2:


This image emphasizes some general periodicity for heights h (modulo 1).
In the following picture the range for h is increased. We have 0<=h<=4. The vertical lines of convergent sequences at integer h=0,1,2 are easily recognizable. The integer heights at h=3,4 do not occur as lines; the surrounding oscillations obviously occur at terms with higher index k>127.
Plot 3:


But we can recognize another topology of brightness now in the left bottom square. See the enhancement of this structure in the next picture.
In the following picture the brightnessparameter for each pixel was logarithmized (base 10), and then a "relief "filter was applied to the picture which renders somehow "terasses" according to the brightnessvalues. Here darker color of the terasses correlate with higher absolute values in the originial data and on the green line the coefficients tend to zero.
Plot 4:


Heights –2<=h<=2
Here the original coloring scheme is used again. For higher negative h's the oscillating and diverging structure of coefficients continue and even seem to improve. The light green line is the zerocoefficientsseries for h=0. (Unfortunately the program read the zeros not correctly, so assigned maximal negative values to them the color of this line should be black, sorry.)
Plot 5: Coefficients A of powerseries


Accidentally I applied a "fun"coloringscheme – but the result is artistically impressive. It looks like the "hellgate of tetrationpowerseries" – an extremely good metaphor! (Don't get lost in the chaos of that powerseries  you're now warned ...)
Plot 6: "The hellgate of tetrationpowerseries"


Here we get wildly diverging entries, where we even don't find the larger oscillating structure of the diverging entries at fractional heights.
Since the a_{k} are computed by a formula, such that in the denominators occur products of
(u1)(u^{2}1)(u^{3}1)...(u^{k}1)
where u=log(t), (see [Helms081],pg 21) and is here about 1.098, we have denominators near zero, which lets us expect hyperbolic effects. Indeed I could not find a larger oscillating structure is I got this in the example (using t=exp(2)) in the first 128 terms.
Graph 2:


The detail at small indexes k looks as follows
Graph 3:


Using base t=2 we seem to get convergent sequences of a_{k}.so the fractionally iterated functions U_{2}^{oh}(x) have convergenceradius of at least 1. However, for fractional iterates the sequences of coefficients have still an interesting largerscale oscillating behaviour.
Graph 4:


The problem with these powerseries is that although the sequences of coefficients converge to zero and thus have a radius of convergence of at least 1, the functions of fractional heights are not well summable (Cesaro and/or Eulersummation) and convergence cannot much be accelerated by these methods, if at all.
I possibly got an extremely simple solution for the problem of alternating divergence for fractional iteration.
We can transform the coefficients a_{k} of a powerseries for a fractional or integer iterate of U_{t}^{oh}(x) in the following way.
Assume the direct computation. In matrixnotation we would collect all a_{k} in a columnvector A and then evaluate U_{t}^{oh}(x) by the dotproduct
(1) V(x)~ * A = U_{t}^{oh}(x)
(see matrixdefinitions at appendix 4.1). But we may insert a transformation by the factoriallyscaled Stirlingmatrix 2^{nd} kind U. This matrix is the operator for transforming V(x)>V(e^{x} 1) using base exp(1):
V(x)~ * U = V(e^{x} 1)~
Obviously, if we use x' = log(1+x) we have
V(x')~ * U = V(x)~
So we may rewrite (1) as
(V(x')~ * U ) * A = U_{t}^{oh}(x)
and use associativity to change order of evaluation
V(x')~
* (U * A) = U_{t}^{oh}(x)
V(x')~ * A'
= U_{t}^{oh}(x)
We may call the expression in the parenthese A' , which is now a sort of "Stirlingtransform" of A, and we observe, that the coefficients in A' form an ordinary convergent sequence.
We have then
a'_{k} 
= 
_{} 
where S2_{k,j} is the Stirlingnumber 2^{nd} kind of row k, column j of the lower triangular Stirlingmatrix 
U_{t}^{oh}(x) 
= 
_{} 


= 
_{} 

(I checked this only for the base t=exp(2) so far)
Here is the graph of the sequences of coefficients for some fractional heights (Sorry, I used my Pari/GPnotation fS2F for U here)
Graph 5:


The graph suggests, that for all heights h>2 (as far as tested), including the fractional ones, the sequences of coefficients converge nicely.
This transformation has another benefit: since x' = log(1+x) which is smaller than x this extends remarkably the range for the xparameter. Possibly we have now functions, which converge for any real x.
To see this explicitely written as powerseries:
The raw computation of the powerseriesexpansion for U_{t}^{o0.5}(x) is:
U_{t}^{o0.5}(x)
= 1.4142*x + 0.58579*x^2 + 0.085506*x^3 + 0.0043325*x^4 + 0.00052932*x^5 
0.00034288*x^6
+ ...
+ 0.000054370*x^33  0.000035807*x^34 + 0.000010332*x^35
+ 0.000022808*x^36  0.000064328*x^37 + 0.00011487*x^38
+...
+ 0.33002*x^87  0.23926*x^88 + 0.093113*x^89 + 0.12710*x^90  0.44514*x^91 + 0.89110*x^92
 1.5028*x^93 + 2.3278*x^94  3.4251*x^95 + 4.8684*x^96  6.7486*x^97 +
9.1779*x^98  12.294*x^99 + 16.266*x^100  21.300*x^101 + 27.647*x^102 
35.611*x^103 + 45.558*x^104  57.932*x^105 + 73.264*x^106  92.190*x^107 +
115.47*x^108  144.00*x^109 + 178.84*x^110  221.26*x^111 + 272.73*x^112 
334.95*x^113 + 409.94*x^114  499.98*x^115 + 607.73*x^116  736.19*x^117 +
888.75*x^118  1069.2*x^119 + 1281.7*x^120  1530.7*x^121 + 1821.2*x^122 
2158.0*x^123 + 2546.2*x^124  2990.3*x^125 + 3494.4*x^126  4061.1*x^127 +
O(x^128)
which shows divergence of the coefficients, which will increase to a hypergeometric rate. The smallest coefficient occurs at x^35 (red marked) thus limits in principle the achievable accuracy.
The Stirlingtransformed series for U_{t}', which means we have also to insert x'=log(1+x) for x, looks like
U_{t}' ^{o0.5}(x)
= 1.4142*x + 1.2929*x^2 + 0.90699*x^3 + 0.53323*x^4 + 0.27431*x^5 + 0.12690*x^6
+ 0.053781*x^7 + 0.021155*x^8 + 0.0078010*x^9 + 0.0027175*x^10 +
0.00089981*x^11 + 0.00028462*x^12 + 0.000086358*x^13 + 0.000025224*x^14 +
0.0000071125*x^15
+...
+ 1.3418e15*x^33  1.8094e16*x^34  2.1937e16*x^35
+ 3.0532e16*x^36  2.7140e16*x^37 + 2.0632e16*x^38
+...
+4.9450e28*x^87  9.3801e29*x^88  6.8606e29*x^89 + 1.1767e28*x^90 
1.1725e28*x^91 + 9.8682e29*x^92  7.6343e29*x^93 + 5.6126e29*x^94 
3.9868e29*x^95 + 2.7622e29*x^96  1.8778e29*x^97 + 1.2574e29*x^98 
8.3165e30*x^99 + 5.4443e30*x^100  3.5328e30*x^101 + 2.2749e30*x^102 
1.4550e30*x^103 + 9.2502e31*x^104  5.8486e31*x^105 + 3.6794e31*x^106 
2.3041e31*x^107 + 1.4366e31*x^108  8.9205e32*x^109 + 5.5178e32*x^110 
3.4004e32*x^111 + 2.0880e32*x^112  1.2776e32*x^113 + 7.7910e33*x^114 
4.7348e33*x^115 + 2.8676e33*x^116  1.7308e33*x^117 + 1.0411e33*x^118 
6.2394e34*x^119 + 3.7257e34*x^120  2.2162e34*x^121 + 1.3129e34*x^122 
7.7448e35*x^123 + 4.5475e35*x^124  2.6568e35*x^125 + 1.5437e35*x^126 
8.9147e36*x^127 + O(x^128)
with nicely diminuishing coefficients from the beginning (and even alternating).
Using x=1, while U_{t}^{o0.5}(x) has to be evaluated by means of a divergent summation technique, we get the same result by ordinary sum of the powerseries for U_{t}' ^{o0.5} using x'=log(1+x)~0.69315 which even allows to neglect all terms from, say k=35, and we will still get a result accurate to an error smaller than something like 1e50.
The bitmap for 161 fractional iterates between –2<h<2 shows that nice behave in a greater overview. Here the dark area at the bottom indicates coefficients tending to zero.
Plot 7: Coefficients A' of powerseries Stirlingtransformed of order 1; base t=exp(2)


This bitmap suggests very strongly, that indeed we can expect convergent sequences of coefficients in the powerseries for fractional iterates.
Since the powerseries for this base are "more diverging" I tried powers of U.
Bitmap by the simple Stirlingtransform using U as before
Plot 8: Coefficients A' of powerseries Stirlingtransformed of order 1 , base t=exp(3)


The transformation works as expected.
The matrixformula was:
(2) V(x)~ * A = U_{t}^{oh}(x) //restated
Using the second power of U we have to start with:
V(x)~ * U^{2} = V(exp(e^{x} 1)1)~
Now, if we use x" = log(1+log(1+x)) we have
V(x")~ * U^{2} = V(x)~
So we may rewrite (2) as
(V(x")~ * U^{2} ) * A = U_{t}^{oh}(x)
and use associativity to change order of evaluation
V(x")~
* (U^{2} * A) = U_{t}^{oh}(x)
V(x")~ * A" = U_{t}^{oh}(x)
We can adapt the formula in serial notation accordingly.
We get the following bitmap:
Plot 9: Coefficients A" of powerseries Stirlingtransformed of order 2 , base t=exp(3)


The matrixformula was:
(3) V(x)~ * A = U_{t}^{oh}(x) //restated
Using the third power of U we have to start with:
V(x)~ * U^{3} = V(exp(exp(e^{x} 1)1)1)~
Now, if we use x'" = log(1+log(1+log(1+x))) we have
V(x'")~ * U^{3} = V(x)~
So we may rewrite (3) as
(V(x'")~ * U^{3} ) * A = U_{t}^{oh}(x)
and use associativity to change order of evaluation
V(x'")~
* (U^{3} * A) = U_{t}^{oh}(x)
V(x'")~ * A'" = U_{t}^{oh}(x)
We can adapt the formula in serial notation accordingly.
We get the following bitmap:
Plot 10: Coefficients A'" of powerseries Stirlingtransformed of order 3 , base t=exp(3)


Although now no black areas occur (the coefficients do not converge fast enough to zero), we can observe, that the darkening of the area read from left expands faster than the occurence of alternating signs. That may suggest, that we will asymptotically still get convergent series of coefficients, which means with convergenceradius of the function of at least 1.
Analoguously to the classical Eulertransform in other cases of evaluating alternating divergent series, we can apply higher "orders" of the transformation. What we see is (and what are the hypotheses so far) with higher orders:
a) the increasing size of the structures
b) that the alternating signs disappear to the left
Here is a table of results, which is based on a slightly different implementation of this "Stirlingtransformation". I use the conventional path via partial sums
_{}
first; setting x=1. These partial sums are then "Stirlingtransformed" (with norming)
Let, for a row r denote the rowsum of U as S_{r}
_{}
(where again, S2_{r,c} are the Stirlingnumbers 2^{nd} kind) then the transforms of the partial sums are
_{}
and this gives then in the limit
_{}
In matrixnotation this is
PS
= DR * A // where DR is the
triangular unitmatrix
//
which implements partial summing
U_{t}^{oh}(x)
= lim _{r>inf} rownorm(U)*PS [r]
//
where r indicates the rowindex and rownorm norms
//a
row, such that the rowsum equals 1
The advantage of the transformation is, as said above, that we get convergent series, and – for the tested example – high accuracy with 128 terms.
What I got by this, using base t=exp(2) is the following table for heights –2<=h<=2 (see full table at appendix)
h 
U_{t}^{oh}(1) ,128 terms used 
difference of results 
Integer
iterates, 
2.000 
0.148781642394 
4.09201648025E61 
0.148781642394 
1.975 
0.151764471472 
1.00757696730E55 

1.950 
0.154814688293 
2.20253988524E55 

1.925 
0.157934114668 
3.53513519222E55 

1.900 
0.161124633560 
4.95458163457E55 

1.875 
0.164388191607 
6.41053955877E55 

1.850 
0.167726801784 
7.85448279957E55 

... 
... 
... 

1.025 
0.338740003829 
4.52893426113E56 

1.000 
0.346573590280 
5.53348935396E123 
0.346573590280 
0.9750 
0.354630797379 
5.13722405573E56 

... 
... 


0.4500 
1.91924350162 
1.75550341076E55 

0.4750 
2.00205867213 
1.60763823499E55 

0.5000 
2.09017531971 
1.44447235835E55 

0.5250 
2.18405909158 
1.27032035841E55 

0.5500 
2.28422767154 
1.08958493793E55 

... 
... 
... 

0.9500 
5.54223517697 
1.54985526435E56 

0.9750 
5.94359516189 
8.24955710752E57 

1.000 
6.38905609893 
2.42197375312E67 
6.38905609893 
1.025 
6.88513866689 
8.97563144073E57 

1.050 
7.43957118855 
1.83978085590E56 

... 
... 
... 

1.450 
45.4551347734 
2.78647933160E34 

1.475 
53.8234128366 
1.14244842556E32 

1.500 
64.3887770782 
4.49621086353E31 

1.525 
77.8950259312 
1.70109100327E29 

1.550 
95.3950973612 
6.19568773605E28 

... 
... 
... 

1.950 
65150.4833093 
0.0000850527607210 

1.975 
145391.211783 
0.00190956027798 

2.000 
354374.278789 
0.0419734968018 
354374.440984 
From the matrixbased formulae the effect can nicely be seen.
Since, by the matrixapproach,
V(x)~ U_{t}^{h} = V(U_{t}°^{h}(x))~
we have by this transformation, which I expand here to a full similaritytransformation:
(V(log(1+x))~
*U )* U_{t}^{h} = V(U_{t}°^{h}(x))~
(V(log(1+x))~ *(U * U_{t}^{h}
*U^{1} ) = V(U_{t}°^{h}(x))~
*U^{1}
= V(log(1+U_{t}°^{h}(x)))~
U * U_{t}^{h}
*U^{1} = U * (^{d}V(u) U)^{h} *U^{1}
= (^{d}V(1/u)
^{d}V(u))* U * (^{d}V(u) U)^{h1} * ^{d}V(u)
U*U^{1}
= ^{d}V(1/u) (^{d}V(u)* U * (^{d}V(u)
U)^{h1} )* ^{d}V(u)
= ^{d}V(1/u) (^{d}V(u) U)^{h}*
^{d}V(u)
(1) = ^{d}V(1/u) * U_{t}^{h} * ^{d}V(u)
which also results in the transformation
V(log(1+x)/u)~ * U_{t}^{h} = V(log(1+U_{t}°^{h}(x))/u)~
The effect of (1) is, that in the symbolic powerseries we remove just powers of u. Recall the the symbolic powerseries for U_{t}°^{h}(x):
U_{t}°^{h}(x) = _{}
where a_{u,k}(u^{h}) are polynomials in u and u^{h} with coeffients depending on k, we have also the factor u_{k}. This u_{k} factor disappears by the similaritytransformation in the following way
U_{t}°^{h}(x) = _{}
But since this is only the removal of a factor with geometric increase, but the powerseries seem to have convergenceradius zero (hypothetically extending I.N.Baker's result for exp(x)1), we are not yet definitely on the successide. However, since many more terms of the powerseries are decreasing in absolute value, we should be able to get better approximations this way, at least for bases t, where u=log(t)>1 .
(to be continued)
The difficult to handle powerseries for fractional iteration seem to be transformable into convergent series by a simple transform using the Stirlingnumbers 2^{nd} kind.
Gottfried Helms, 24.07.2008
All derivations are done in the matrixnotation; however can be reexpressed in common serial notation. The matrixnotation is simply meant to keep the formulae concise.
All matrices are assumed as of infinite size (to resemble the infinite terms of powerseries); in the current computations I truncated them to size n=128 resp nxn=128x128.
The used matrixoperators for the Utetration are named U resp. U_{t} and the other involved standardmatrices are as follows:
V(x) :=
column_{r=0..inf}[1,x,x^{2},x^{3},...x^{r},...]
an infinite
"vandermonde" (column) vector of a general x
V(x)~ := the transpose ; the symbol is taken from the convention in
Pari/GP
^{d}V(x) := the diagonal arrangement of V(x)
The column and rowindices are beginning at zero
VZ :=
matrix_{r=0..inf,c=0..inf}[c^{r}] 

P
:= matrix_{r=0..inf,c=0..inf}[binomial(r,c)]


^{d}F := diag(0!,1!,2!,...) 

S2 := matrix_{r=0..inf,c=0..inf}[s2_{r,c}]


S1 := matrix_{r=0..inf,c=0..inf}[s1_{r,c}]


U = fS2F:= matrix_{r=0..inf,c=0..inf}[s2_{r,c}
c!/r!] 

u :=log(t) // baseparameter 

DR := matrix_{r=0..inf,c=0..inf}[if(r>=c,1)] 

The function U_{t}°^{h}(x) can be computed by the dotproduct of V(x) and column 1 of the h^{th}power of U_{t} (which contains the required coefficients of the powerseries)
V(x)~
* U_{t}^{h} = V(U_{t}°^{h}(x) )~
or, referencing the second column of U_{t} only:
V(x)~ * U_{t}^{h}
[,1] = U_{t}°^{h}(x)
The fractional powers of the matrix U_{t} is computed by diagonalization:
U_{t} = W^{1} * D * W // where the diagonal D equals ^{d}V(u) and W is triangular
This can exactly be solved for any finite dimension where the entries are constant wrt to selected matrixsize. (A simple recursive solution). Then the fractional h'th powers of U_{t} are computed by
U_{t}^{h} = W^{1} * ^{d}V(u^{h}) * W
Important: for u=1, t=exp(1) (and bases near that value) the matrixlogarithm is applied instead.
Table of approximates of U_{t}^{oh}(1) for h=2..2 in 160 steps using base t=exp(2)~6.38905609893
h 
U_{t}^{oh}(1) 
difference of results 
2.000 
0.148781642394 
4.09201648025E61 
1.975 
0.151764471472 
1.00757696730E55 
1.950 
0.154814688293 
2.20253988524E55 
1.925 
0.157934114668 
3.53513519222E55 
1.900 
0.161124633560 
4.95458163457E55 
1.875 
0.164388191607 
6.41053955877E55 
1.850 
0.167726801784 
7.85448279957E55 
1.825 
0.171142546188 
9.24093772260E55 
1.800 
0.174637578968 
1.05285638946E54 
1.775 
0.178214129401 
1.16810569215E54 
1.750 
0.181874505127 
1.26678601269E54 
1.725 
0.185621095548 
1.34646777765E54 
1.700 
0.189456375403 
1.40537883466E54 
1.675 
0.193382908532 
1.44241617525E54 
1.650 
0.197403351836 
1.45713893941E54 
1.625 
0.201520459451 
1.44974402527E54 
1.600 
0.205737087133 
1.42102600088E54 
1.575 
0.210056196899 
1.37232332072E54 
1.550 
0.214480861902 
1.30545308365E54 
1.525 
0.219014271584 
1.22263673030E54 
1.500 
0.223659737106 
1.12641916825E54 
1.475 
0.228420697091 
1.01958383306E54 
1.450 
0.233300723675 
9.05066148888E55 
1.425 
0.238303528916 
7.85867746245E55 
1.400 
0.243432971557 
6.64973635962E55 
1.375 
0.248693064184 
5.45274331996E55 
1.350 
0.254087980802 
4.29494671718E55 
1.325 
0.259622064847 
3.20130807787E55 
1.300 
0.265299837682 
2.19396550105E55 
1.275 
0.271126007591 
1.29179928123E55 
1.250 
0.277105479321 
5.10105315764E56 
1.225 
0.283243364195 
1.39621203423E56 
1.200 
0.289544990856 
6.49792278030E56 
1.175 
0.296015916665 
1.01672701607E55 
1.150 
0.302661939812 
1.24055075057E55 
1.125 
0.309489112197 
1.32500139054E55 
1.100 
0.316503753120 
1.27715294403E55 
1.075 
0.323712463854 
1.10706770668E55 
1.050 
0.331122143163 
8.27389786305E56 
1.025 
0.338740003829 
4.52893426113E56 
1.000 
0.346573590280 
5.53348935396E123 
0.9750 
0.354630797379 
5.13722405573E56 
0.9500 
0.362919890491 
1.07006321730E55 
0.9250 
0.371449526905 
1.65066443311E55 
0.9000 
0.380228778734 
2.23749445643E55 
0.8750 
0.389267157395 
2.81328981940E55 
0.8500 
0.398574639809 
3.36195528409E55 
0.8250 
0.408161696458 
3.86891434953E55 
0.8000 
0.418039321439 
4.32140384738E55 
0.7750 
0.428219064697 
4.70870803363E55 
0.7500 
0.438713066612 
5.02232933208E55 
0.7250 
0.449534095129 
5.25609461431E55 
0.7000 
0.460695585670 
5.40619756770E55 
0.6750 
0.472211684045 
5.47117927292E55 
0.6500 
0.484097292638 
5.45185054828E55 
0.6250 
0.496368120150 
5.35116089595E55 
0.6000 
0.509040735221 
5.17401997699E55 
0.5750 
0.522132624272 
4.92707843411E55 
0.5500 
0.535662253964 
4.61847555857E55 
0.5250 
0.549649138683 
4.25756175589E55 
0.5000 
0.564113913536 
3.85460400264E55 
0.4750 
0.579078413362 
3.42048250880E55 
0.4500 
0.594565758348 
2.96638661656E55 
0.4250 
0.610600446866 
2.50351759218E55 
0.4000 
0.627208456258 
2.04280541998E55 
0.3750 
0.644417352338 
1.59464601053E55 
0.3500 
0.662256408490 
1.16866441150E55 
0.3250 
0.680756735328 
7.73508687414E56 
0.3000 
0.699951422006 
4.16678141471E56 
0.2750 
0.719875690386 
1.04388516000E56 
0.2500 
0.740567063404 
1.58524242474E56 
0.2250 
0.762065549163 
3.68661106455E56 
0.2000 
0.784413842431 
5.24078433084E56 
0.1750 
0.807657545455 
6.24269166760E56 
0.1500 
0.831845410221 
6.70109033021E56 
0.1250 
0.857029604574 
6.63770739168E56 
0.1000 
0.883266004898 
6.08609846115E56 
0.07500 
0.910614518426 
5.09026502228E56 
0.0500 
0.939139438629 
3.70307626659E56 
0.0250 
0.968909837620 
1.98454393587E56 
0.0000 
1.00000000000 
2.041281526E202 
0.0250 
1.03248990321 
2.18177298377E56 
0.0500 
1.06646575014 
4.49041019351E56 
0.0750 
1.10202056051 
6.85578525869E56 
0.1000 
1.13925482862 
9.20974045399E56 
0.1250 
1.17827725587 
1.14876877214E55 
0.1500 
1.21920556807 
1.36300468824E55 
0.1750 
1.26216742863 
1.55834930026E55 
0.2000 
1.30730146082 
1.73019908921E55 
0.2250 
1.35475839393 
1.87476010084E55 
0.2500 
1.40470235077 
1.98910473981E55 
0.2750 
1.45731229648 
2.07120446227E55 
0.3000 
1.51278367212 
2.11993867020E55 
0.3250 
1.57133023995 
2.13508068412E55 
0.3500 
1.63318617222 
2.11726219652E55 
0.3750 
1.69860842041 
2.06791807521E55 
0.4000 
1.76787940830 
1.98921378529E55 
0.4250 
1.84131009978 
1.88395802419E55 
0.4500 
1.91924350162 
1.75550341076E55 
0.4750 
2.00205867213 
1.60763823499E55 
0.5000 
2.09017531971 
1.44447235835E55 
0.5250 
2.18405909158 
1.27032035841E55 
0.5500 
2.28422767154 
1.08958493793E55 
0.5750 
2.39125782952 
9.06643473920E56 
0.6000 
2.50579359389 
7.25740373036E56 
0.6250 
2.62855575211 
5.50887633339E56 
0.6500 
2.76035292858 
3.85775699610E56 
0.6750 
2.90209454058 
2.33696348512E56 
0.7000 
3.05480599903 
9.74789623601E57 
0.7250 
3.21964660129 
2.05588436331E57 
0.7500 
3.39793066465 
1.18645675450E56 
0.7750 
3.59115257551 
1.95575725626E56 
0.8000 
3.80101658899 
2.50713260877E56 
0.8250 
4.02947241486 
2.83982801087E56 
0.8500 
4.27875788211 
2.95845886162E56 
0.8750 
4.55145030117 
2.87265617216E56 
0.9000 
4.85052856291 
2.59660405743E56 
0.9250 
5.17944855466 
2.14848564740E56 
0.9500 
5.54223517697 
1.54985526435E56 
0.9750 
5.94359516189 
8.24955710752E57 
1.000 
6.38905609893 
2.42197375312E67 
1.025 
6.88513866689 
8.97563144073E57 
1.050 
7.43957118855 
1.83978085590E56 
1.075 
8.06155846276 
2.79887303842E56 
1.100 
8.76212065682 
3.74458219387E56 
1.125 
9.55452324610 
4.34775784184E56 
1.150 
10.4548261171 
2.06904925500E55 
1.175 
11.4825897977 
2.03275257844E53 
1.200 
12.6617904993 
1.48421551843E51 
1.225 
14.0220149408 
1.01379756997E49 
1.250 
15.6000333024 
6.51488362789E48 
1.275 
17.4418878886 
3.94903550037E46 
1.300 
19.6056919163 
2.26333836768E44 
1.325 
22.1654160710 
1.22932849531E42 
1.350 
25.2160637847 
6.34129229359E41 
1.375 
28.8808211306 
3.11286011567E39 
1.400 
33.3210482169 
1.45695775320E37 
1.425 
38.7504117507 
6.51372560175E36 
1.450 
45.4551347734 
2.78647933160E34 
1.475 
53.8234128366 
1.14244842556E32 
1.500 
64.3887770782 
4.49621086353E31 
1.525 
77.8950259312 
1.70109100327E29 
1.550 
95.3950973612 
6.19568773605E28 
1.575 
118.404353175 
2.17527216428E26 
1.600 
149.142852032 
7.37145701535E25 
1.625 
190.926311527 
2.41399561943E23 
1.650 
248.811306070 
7.64832842296E22 
1.675 
330.686113737 
2.34705994831E20 
1.700 
449.164016764 
6.98342783166E19 
1.725 
624.964213535 
2.01668882457E17 
1.750 
893.139076207 
5.65789354699E16 
1.775 
1314.93820096 
1.54354082962E14 
1.800 
2001.26271589 
4.09839428311E13 
1.825 
3160.95197578 
1.06001083639E11 
1.850 
5204.73282794 
0.000000000267276805071 
1.875 
8980.30615555 
0.00000000657514974489 
1.900 
16333.8660823 
0.000000157931503896 
1.925 
31535.3809448 
0.00000370649148228 
1.950 
65150.4833093 
0.0000850527607210 
1.975 
145391.211783 
0.00190956027798 
2.000 
354374.278789 
0.0419734968018 
[Bak] Baker,
Irvine Noel;
Zusammensetzungen ganzer Funktionen
1958; Mathematische Zeitschrift, Vol 69, Pg 121163,
[Vol] Volkov, I.I.;
Riesz summation method
Online reference Springer:Encyclopaedia of Mathematics
Edited by Michiel Hazewinkel
online at http://eom.springer.de/r/r082300.htm
[Helms,0711] Helms, Gottfried;
Coefficients for fractional iterations of the Ufunction (first version)
online at http://go.helmsnet.de/math/tetdocs/CoefficientsForUTetration.htm
[Helms,0712] Helms, Gottfried;
Coefficients for fractional iterations of the Ufunction (extended version)
online at http://go.helmsnet.de/math/tetdocs/htmltable_utetrationFractionalIteration.htm
[Helms,081] Helms, Gottfried;
Continuous functional iteration
pg 21, online at http://go.helmsnet.de/math/tetdocs/ContinuousfunctionalIteration.pdf