Hi Sheldon -
I've fiddled a bit with the problem of rational periods to see how far, and why it differs from the irrational-period orbits. I don't know whether this is of any use at all, at least it has been an interesting exercise.
For an example let's take the period of 4, so the parameter phi=Pi/2, then u=cos(phi)+I*sin(phi)=1*I, t=exp(u)=e^I, b=exp(u/t)=e^(I/e^I). Let's call our base-function f(x)=b^x, and if h-fold iterated f(x,h).
I use a slightly different shift for the recentering of the series for the regular-tetration base-function, but found it convenient in my tetration-analyses: x'= (x+1)*t and x"=x/t -1 so g(x,h) = f(x',h)" and conversely f(x,h)=g(x",h)' .
The Bell-matrix for g(x) = f(x')" = (b^x')" = b^((x+1)*t)/t-1 is triangular and just the matrix of Stirlingnumbers 2nd kind rowwise rescaled by powers of u and reciprocals of factorials (its generating function g(z)=exp(u*z)-1 ).
However, having repeated eigenvalues here the standard computation of diagonalization fails, so we cannot compute the schroeder-function which is needed for the regular superfunction.
On the other hand, when the diagonal has all units then we have the alternative of the matrix-logarithm to define a formal (bivariate) powerseries for the superfunction in terms of x and the height-parameter h. But although the powers of u on the diagonal have absolute values of the unit, it is again not possible to do the matrix-logarithm. (If u were a complex unit-root of *irrational* order, then all integer powers of u, and thus eigenvalues, are different, especially different from 1, and the diagonalization can be used to build up the matrices/formal powerseries for regular iteration)
So looking again at the formal expression for the superfunction in x and h (by diagonalization, and where the parameters u and h are still kept symbolic and not yet evaluated) we find, that in the denominators occur expressions (1-u^4), (1-u^8), (1-u^12) and so forth (which is well known). These become zero if evaluated and produce singularities.
Here I remembered an analoguous case: if we remove singularities from the zeta-function zeta*(s)=zeta(s)-1/(s-1) we get an entire function, and the same is possible with the gamma-function as well; if we remove all the singularities by a similar scheme, we get the "incomplete gamma" which is again entire, and nicely converging (and has also apparently interesting and useful properties)
So I looked what it would give, if I remove the singularites here as well, thus creating a somehow "incomplete" exponentiation/tetration g*(x) resp g*(x,h).
I cancelled all occurences of (1-u^4) in the denominators and could now evaluate the formal powerseries for our selected parameter u=1*I. So it is possible to define a schroederfunction and also a superfunction, and we talk about g*(x,h) and f*(x,h)=g*(x",h)' in the following.
Because the explicite schroeder-function is only needed intermediately I do not show the coefficients here. More concise is the matrix POLY of coefficients, which have to be premultiplied by the consecutive powers of the x-parameter of g*(x,h) (or let's say z0 instead of x because we use it as complex-valued initial value b^^0 for the iteration) and postmultiplied by consecutive powers of u^h .
Here the singularities are removed in the way, that at the coefficient z0^{4k+1},z0^{4k+2},z0^{4k+3},z0^{4k+4} in the denominator the (1-u^4)^k was cancelled. Reconstruction of the original g(x,s) by this means to divide the rows of POLY by that multiplicities of zeros (but I'm not good in application and explanation of the L'Hospital-rule here, and under which circumstances I could make use of it for our case here)
Now in our case it also comes out to be more convenient to evaluate the postmultiplication first, so I got the formal powerseries for g*(x,h) for the first few h: g*(x,0) = x, g*(x,1), g*(x,2) , g*(x,3), g*(x,4), g*(x,5)=g*(x,1) where it becomes periodic. Also only the first powers up to x^4 have non-null coefficients:
The coefficients-matrix POLY allows fractional heights, however taking the complex u to fractional powers is not unambiguous, so the following is just for an impression:
The function g*(x,h) can also be converted from the expression using u^(k*h) here, which is Dirichlet-like in terms of h, into a taylor-series in terms of that parameter; we get then another version of the matrix POLY. I can send this coefficients later.
So, what is this all good for? Hmm, really don't know, I'd say it reduces to a nice exercise. Perhaps it gives another impulse for the comparision with the whereabouts of the bases b which allow periods of irrational length...
Gottfried
I've fiddled a bit with the problem of rational periods to see how far, and why it differs from the irrational-period orbits. I don't know whether this is of any use at all, at least it has been an interesting exercise.
For an example let's take the period of 4, so the parameter phi=Pi/2, then u=cos(phi)+I*sin(phi)=1*I, t=exp(u)=e^I, b=exp(u/t)=e^(I/e^I). Let's call our base-function f(x)=b^x, and if h-fold iterated f(x,h).
I use a slightly different shift for the recentering of the series for the regular-tetration base-function, but found it convenient in my tetration-analyses: x'= (x+1)*t and x"=x/t -1 so g(x,h) = f(x',h)" and conversely f(x,h)=g(x",h)' .
The Bell-matrix for g(x) = f(x')" = (b^x')" = b^((x+1)*t)/t-1 is triangular and just the matrix of Stirlingnumbers 2nd kind rowwise rescaled by powers of u and reciprocals of factorials (its generating function g(z)=exp(u*z)-1 ).
However, having repeated eigenvalues here the standard computation of diagonalization fails, so we cannot compute the schroeder-function which is needed for the regular superfunction.
On the other hand, when the diagonal has all units then we have the alternative of the matrix-logarithm to define a formal (bivariate) powerseries for the superfunction in terms of x and the height-parameter h. But although the powers of u on the diagonal have absolute values of the unit, it is again not possible to do the matrix-logarithm. (If u were a complex unit-root of *irrational* order, then all integer powers of u, and thus eigenvalues, are different, especially different from 1, and the diagonalization can be used to build up the matrices/formal powerseries for regular iteration)
So looking again at the formal expression for the superfunction in x and h (by diagonalization, and where the parameters u and h are still kept symbolic and not yet evaluated) we find, that in the denominators occur expressions (1-u^4), (1-u^8), (1-u^12) and so forth (which is well known). These become zero if evaluated and produce singularities.
Here I remembered an analoguous case: if we remove singularities from the zeta-function zeta*(s)=zeta(s)-1/(s-1) we get an entire function, and the same is possible with the gamma-function as well; if we remove all the singularities by a similar scheme, we get the "incomplete gamma" which is again entire, and nicely converging (and has also apparently interesting and useful properties)
So I looked what it would give, if I remove the singularites here as well, thus creating a somehow "incomplete" exponentiation/tetration g*(x) resp g*(x,h).
I cancelled all occurences of (1-u^4) in the denominators and could now evaluate the formal powerseries for our selected parameter u=1*I. So it is possible to define a schroederfunction and also a superfunction, and we talk about g*(x,h) and f*(x,h)=g*(x",h)' in the following.
Because the explicite schroeder-function is only needed intermediately I do not show the coefficients here. More concise is the matrix POLY of coefficients, which have to be premultiplied by the consecutive powers of the x-parameter of g*(x,h) (or let's say z0 instead of x because we use it as complex-valued initial value b^^0 for the iteration) and postmultiplied by consecutive powers of u^h .
Code:
Poly
0 u^(0h)* u^(1h)* u^(2h)* u^(3h)* u^(4h)* u^(5h)* u^(6h)* u^(7h)*
-------------------------------------------------------------------------------------------------------------------------
(z0/2)^0/0!)* . . . . . . . .
(z0/2)^1/1!)* . 2 . . . . . .
(z0/2)^2/2!)* . -2+2*I 2-2*I . . . . .
(z0/2)^3/3!)* . 2-6*I 12*I -2-6*I . . . .
(z0/2)^4/4!)* . 16*I -28-44*I 48+24*I -20+4*I . . .
(z0/2)^5/5!)* . -8-40*I . . . 8+40*I . .
(z0/2)^6/6!)* . 720+480*I -288-192*I . . -720-480*I 288+192*I .
(z0/2)^7/7!)* . -12880+1120*I 10080-2016*I -2352+1344*I . 12880-1120*I -10080+2016*I 2352-1344*I
... ... ... ... ... ... ... ... ...
-------------------------------------------------------------------------------------------------------------------------Here the singularities are removed in the way, that at the coefficient z0^{4k+1},z0^{4k+2},z0^{4k+3},z0^{4k+4} in the denominator the (1-u^4)^k was cancelled. Reconstruction of the original g(x,s) by this means to divide the rows of POLY by that multiplicities of zeros (but I'm not good in application and explanation of the L'Hospital-rule here, and under which circumstances I could make use of it for our case here)
Now in our case it also comes out to be more convenient to evaluate the postmultiplication first, so I got the formal powerseries for g*(x,h) for the first few h: g*(x,0) = x, g*(x,1), g*(x,2) , g*(x,3), g*(x,4), g*(x,5)=g*(x,1) where it becomes periodic. Also only the first powers up to x^4 have non-null coefficients:
Code:
Integer heights ; periodic; only the coefficients up to z0^4 are not null
0 h=0 h=1 h=2 h=3 h=4 h=5 ...
---------+----+-----+------+-----+----+-----+----
z0^0/0!* . . . . . . ...
z0^1/1!* 1 I -1 -I 1 I ...
z0^2/2!* . -1 1-I I . -1
z0^3/3!* . -I 3*I -2*I . -I
z0^4/4!* . 1 -6-5*I 6*I . 1
z0^5/5!* . . . . . .
... ... ... ... ... ... ... ...
---------+----+-----+------+-----+----+-----+----
example: g*(z0,2) = - z0 + (1-I)/2*z0^2 - 3 I/6*z0^3 - (6+5I)/24*z0^4The coefficients-matrix POLY allows fractional heights, however taking the complex u to fractional powers is not unambiguous, so the following is just for an impression:
Code:
Fractional heights (meaningful ?); h=0.5
g*(z0,0.5)=
+0.70710678+0.70710678*I *z0
-0.10355339 . *z0^2
-0.014297740 . *z0^3
0.0046213626-0.0096763770*I *z0^4
0.011785113-0.017677670*I *z0^5
0.015699029+0.024328478*I *z0^6
-0.022230690-0.0026321999*I *z0^7
0.010378519-0.0080516725*I *z0^8
-0.0010434735-0.00042966558*I *z0^9
0.0024291464-0.0017456513*I *z0^10
...The function g*(x,h) can also be converted from the expression using u^(k*h) here, which is Dirichlet-like in terms of h, into a taylor-series in terms of that parameter; we get then another version of the matrix POLY. I can send this coefficients later.
So, what is this all good for? Hmm, really don't know, I'd say it reduces to a nice exercise. Perhaps it gives another impulse for the comparision with the whereabouts of the bases b which allow periods of irrational length...
Gottfried
Gottfried Helms, Kassel
