Here I present three postings in sci.math. It seems, that the method is not well suited for the interpolation to fractional heights (as I hoped it would be). But - perhaps we can find a workaround. On the other hand: it is not needed that many different methods exist, so...
Also Ioannis (Galidakis) reminded me of the entry in mathworld,"powertower", where he already characterized this type of series. (http://mathworld.wolfram.com/PowerTower.html)
Here the current msgs aus sci.math: ( some edits in double-brackets [< >])
. subject: tetration: another family of powerseries for fractional iteration
The 2.nd msg:
I proceeded for the first few terms s2,s3,s4,s5... Things seem to come out uneasy...
Another idea around? Or: can we work differently with the binomial-compositions (I've only basic understanding/knowledge of the generation-function-concept, for instance...)
Also Ioannis (Galidakis) reminded me of the entry in mathworld,"powertower", where he already characterized this type of series. (http://mathworld.wolfram.com/PowerTower.html)
Here the current msgs aus sci.math: ( some edits in double-brackets [< >])
. subject: tetration: another family of powerseries for fractional iteration
Code:
Maybe this is all known; I didn't see it so far. The idea was triggered by
the comments of V Jovovic in the OEIS concerning the below generating functions.
Consider the sequence of functions
T0(x) = 1, T1(x) = exp(x*1), T2(x) = exp(x*exp(x)), T_h(x) = exp(x*T_{h-1}(x)),...
They are also the generation-functions for the following sequence of powerseries:
T0: 1 + 0 + 0 + ....
T1: 1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 ...
T2: 1 + x + 3/2*x^2 + 10/6*x^3 + 41/24*x^4 + 196/120*x^5 + 1057/720*x^6 + 6322/5040*x^7 +...
T3: 1 + x + 3/2*x^2 + 16/6*x^3 + 101/24*x^4 + 756/120*x^5 + 6607/720*x^6 + 160504/5040*x^7 + ...
T4: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1176/120*x^5 + 12847/720*x^6 + 229384/5040*x^7 + ...
T5: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16087/720*x^6 + 257104/5040*x^7 + ...
T6: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T7: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T8: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T9: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
...
Too: 1 + x + 3/2*x^2 + 4^2/3!*x^3 + 5^3/4!*x^4 + 6^4/5!*x^5 + 7^5/6!*x^6 + 8^6/7!*x^7 + ... //limit h->inf
That means, if x = log(b), we have by this
T0(x) = 1
T1(x) = b = b^^1
T2(x) = b^b = b^^2
T3(x) = b^b^b = b^^3
...
Too(x) = ...^b^b = b^^oo
and for the limit h->inf we have with Too(x) the series for the h-function of b: Too(x) = h(b)
which is convergent for |x|<exp(-1)
[<...>]The 2.nd msg:
Code:
> > (Galidakis replies) :
> > However, the recursive expression for the coefficients
> > given in (6) [<in mathworld, G.H.>] does not seem to allow that.
> >
> > If you can find a way to interpolate between those coefficients for non-natural
> > heights using your matrix method AND at the same time you manage to preserve the
> > functional equation F(x + 1) = e^{x*F(x)}, then, by Jove, you've got a nice
> > analytic solution to tetration :-)
Ok, let's give a start. Recall:
T0: 1 + 0 + 0 + ....
T1: 1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 ...
T2: 1 + x + 3/2*x^2 + 10/6*x^3 + 41/24*x^4 + 196/120*x^5 + 1057/720*x^6 + 6322/5040*x^7 +...
T3: 1 + x + 3/2*x^2 + 16/6*x^3 + 101/24*x^4 + 756/120*x^5 + 6607/720*x^6 + 160504/5040*x^7 + ...
T4: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1176/120*x^5 + 12847/720*x^6 + 229384/5040*x^7 + ...
T5: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16087/720*x^6 + 257104/5040*x^7 + ...
T6: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T7: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T8: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
T9: 1 + x + 3/2*x^2 + 16/6*x^3 + 125/24*x^4 + 1296/120*x^5 + 16807/720*x^6 + 262144/5040*x^7 + ...
...
Too: 1 + x + 3/2*x^2 + 4^2/3!*x^3 + 5^3/4!*x^4 + 6^4/5!*x^5 + 7^5/6!*x^6 + 8^6/7!*x^7 + ... //limit h->inf
We want to interpolate for coefficients of T0.5, means between rows T0 and T1.
I'll rewrite the powerseries without the powers of x. And since we do
the binomial composition of coefficients at like powers of x, we compose
the coefficients down a column; so the common denominator(the factorial) of a
column can be omitted for the scheme.
Thus I get for the original coefficients, only rescaled
T0: 1 0 0 0 0 0 0 0 ...
T1: 1 1 1 1 1 1 1 1 ...
T2: 1 1 3 10 41 196 1057 6322
T3: 1 1 3 16 101 756 6607 65794
T4: 1 1 3 16 125 1176 12847 160504
T5: 1 1 3 16 125 1296 16087 229384
T6: 1 1 3 16 125 1296 16807 257104
T7: 1 1 3 16 125 1296 16807 262144
T8: 1 1 3 16 125 1296 16807 262144
T9: 1 1 3 16 125 1296 16807 262144
...
The first binomial-composition along the columns gives
X0: 1 0 0 0 0 0 0 0 ...
X1: 0 1 1 1 1 1 1 1 ...
X2: 0 -1 1 8 39 194 1055 6320
X3: 0 1 -3 -11 -19 171 3439 46831
X4: 0 -1 5 8 -37 -676 -7243 -64744
X5: 0 1 -7 1 105 1021 7357 21589
X6: 0 -1 9 -16 -161 -1026 -3301 67304
X7: 0 1 -11 37 181 631 -3605 -168125
X8: 0 -1 13 -64 -141 104 10961 246224
X9: 0 1 -15 97 17 -999 -16007 -278711
... ...
The second binomial-composition (using h=0.5)
[< Table 5: this will be the reference-table for the composition of coefficients of T05 >]
Y0: 1 0 0 0 0 0 0 ...
Y1: 0 1/2 1/2 1/2 1/2 1/2 1/2 ...
Y2: 0 1/8 -1/8 -1 -39/8 -97/4 -1055/8
Y3: 0 1/16 -3/16 -11/16 -19/16 171/16 3439/16
Y4: 0 5/128 -25/128 -5/16 185/128 845/32 36215/128
Y5: 0 7/256 -49/256 7/256 735/256 7147/256 51499/256
Y6: 0 21/1024 -189/1024 21/64 3381/1024 10773/512 69321/1024
Y7: 0 33/2048 -363/2048 1221/2048 5973/2048 20823/2048 -118965/2048
Y8: 0 429/32768 -5577/32768 429/512 60489/32768 -5577/4096 -4702269/32768
Y9: 0 715/65536 -10725/65536 69355/65536 12155/65536 -714285/65536 -11445005/65536 ...
... ...
----------------------------------------------------------------------------------------------------
sum. s0 s1 s2 s3 ...
====================================================================================================
T0.5: c0 c1 c2 c3 ...
and T0.5(x) = c0 + c1*x + c2*x^2/2! + c3*x^/3! + ...
the interpolated coefficients c0,c1,c2,... for h=0.5 should then be computed by the
column-sums (and finally the rescaling by the omitted factorials).
The partial sums in the columns converge only badly if at all, so let's look,
whether we can find some analytic solution.
The denominators in the rows can be majorized by powers of 4, and all can then be divided by
2, so let's rewrite this
common scaling
Y0: 1/2 0 0 0 0 0 0 0 0 0 *2 /4^0
Y1: 0 1 1 1 1 1 1 1 1 1 *2 /4^1
Y2: 0 1 -1 -8 -39 -194 -1055 -6320 -41391 -293606 *2 /4^2
Y3: 0 2 -6 -22 -38 342 6878 93662 1219314 16331654 *2 /4^3
Y4: 0 5 -25 -40 185 3380 36215 323720 2128445 -5199340 *2 /4^4
Y5: 0 14 -98 14 1470 14294 102998 302246 -9722034 -332756410 *2 /4^5
Y6: 0 42 -378 672 6762 43092 138642 -2826768 -93176118 -1954258068 *2 /4^6
Y7: 0 132 -1452 4884 23892 83292 -475860 -22192500 -463551132 -7659247332 *2 /4^7
Y8: 0 429 -5577 27456 60489 -44616 -4702269 -105630096 -1778712507 -23047084632 *2 /4^8
Y9: 0 1430 -21450 138710 24310 -1428570 -22890010 -398556730 -5760084330 -51266562490 *2 /4^9
... ...
----------------------------------------------------------------------------------------------------
sum. s0 s1 s2 s3 ...
====================================================================================================
T0.5: c0 c1 c2 c3 ...
and T0.5(x) = c0 + c1*x + c2*x^2/2! + c3*x^/3! + ...
Let's look at the columnsums of the table; that sums, divided by the factorial, give the coefficients
c_k for the T0.5(x)-powerseries.
First, s0 = 1, (remember the scaling extracted to the rhs) ,
so c0 = 1
Next, s1. Here we recognize, that the numbers are the catalan-numbers, and, with the
current scaling have the generation-function 1- sqrt(1-z). Since we want to know
simply the sum, we set z=1 and get for the sum
s1 = 1- sqrt(1-1) = 1
so c1 =1
Next, s2. It becomes more difficult. We can add columns s2 and s1 to get a sequence,
which can formally be expressed as the derivative of the sqrt(1 - z)-function, where
possibly we need also a scaling at z, so likely something like
1 - sqrt(1 - a z)'
It looks, as if the series is divergent, too, so we'll have to see, whether this
operation (and the following, which surely are similar) can be justified/make sense
at all.
-----------------I proceeded for the first few terms s2,s3,s4,s5... Things seem to come out uneasy...
Code:
(msg 3)
(...)
Formally composed by derivatives of sqrt(1-z) I get for the series s1,s2,s3,... the following
generating functions
s0: 1
s1: 1 - 1*sqrt(1-z)
s2: 3 - 3*sqrt(1-z) + 2*z*(sqrt(1-z)')
s3: 16 - 16*sqrt(1-z) + 15*z*(sqrt(1-z)') - 3*z^2*(sqrt(1-z)'')
s4: 125 - 125*sqrt(1-z) + 124*z*(sqrt(1-z)') - 42*z^2*(sqrt(1-z)'') + 4*z^3*(sqrt(1-z)''')
s5: 1296 - 1296*sqrt(1-z) + 1295*z*(sqrt(1-z)') - 550*z^2*(sqrt(1-z)'') + 90*z^3*(sqrt(1-z)''') - 5*z^4*(sqrt(1-z)'''')
...
which have to be evaluated at z=1 to give the value for the sums. Now the derivatives have
a vertical asymptote at z=1, so here are infinities everywhere...
Even more obvious, if I expand the derivatives into terms of sqrt(1-z) I get the following
explicite generating functions for the series of s0,s1,s2,...:
s0: 1
s1: 1 - sqrt(1-z) * ( 1 )
s2: 3 - sqrt(1-z)/(1-z)^1* ( 3 - 4/2*z)
s3: 16 - sqrt(1-z)/(1-z)^2* ( 16 - 49/2*z + 31/4*z^2 )
s4: 125 - sqrt(1-z)/(1-z)^3* ( 125 - 626/2*z + 962/4*z^2 - 408/8*z^3 )
s5: 1296 - sqrt(1-z)/(1-z)^4* (1296 - 9073/2*z + 22784/4*z^2 - 23462/8*z^3 + 7561/16*z^4)
where all except the first two grow unboundedly, if z->1
So for that approach: it looks as if we cannot express a half-iterate based on
this type of powerseries. Pity.... Maybe we can find a workaround - change order
of summation or something else, don't have an idea.Another idea around? Or: can we work differently with the binomial-compositions (I've only basic understanding/knowledge of the generation-function-concept, for instance...)
Gottfried Helms, Kassel
