[edit 25.8, minor textual corrections]

Hmm, my thoughts may be amateurish, but I've time for a bit of discussion of it currently.

That, what I called "my matrix method" is nothing else than a concise notation for the manipulations of exponential-

series, which constitute the iterates of x, s^x, s^s^x .

The integer-iterates.

It comes out,that the coefficients for the series, which has to be summed to get the value for

have a complicated structure, but which is anyway iterative with a simple scheme.

It is a multisum, difficult to write, so I indicate it for the first three iterates m=1,m=2,m=3

m=1:

m=2:

m=3:

In m>2 the binomial-term expands to a multinomial-term.

When denoting

then the notation for the general case m>1 can be shortened:

m=arbitrary, integer>0:

(hope I didn't make an index-error).

This formula can also be found in wrong reference removed (update)

Convergence:

Because we know, that powertowers of finite height are computable by

exponential-series we know also, that for the finite case this sum-

expression converges.

This is due the weight, that the factorials in the denominators accumulate.

Bound for base-parameter s:

However, in the case of infinite height we already know,

that log(s) is limited to -e < log(s) < 1/e

Differentability:

In the general formula above we find the parameter for s as n'th power

of its logarithm (divided by the n'th factorial). This simply provides

infinite differentiablity with respect to s

Differentiation with respect to the height-parameter m could be described

when the eigensystem-decomposition is used (see below)

(but may be I misunderstood the ongoing discussion about this topic

completely - I kept myself off this discussion due to lack

of experience of matrix-differentiation :-( )

In all, I think, that this description is undoubtedly the unique definition

for the tetration of integer heights in terms of powerseries in s and x.

------------------------------------------------

For the continuous case my idea was, that it may be determined

either by eigenvalue-decomposition of the involved matrices, or,

where this is impossible due to inadmissible parameters s,

by the matrix-logarithm.

The eigenvalue-decomposition has the diasadvantage, that its

properties are not yet analytically known, while matrix-logarithm

is sometimes an option, if the eigenvalue-decomposition

fails since it seems, that the resulting coefficients could be

easier be determined.

An example is the U-iteration U_s^(m)(x) = s^U_s^(m-1)(x) - 1,

where s=e, which we discuss here as "exp(s)-1" - version. For this

case Daniel already provided an array of coefficients for a series

which allows to determine the U-tetration U_s^(m)(x) for fractional,

real and even complex parameters m.

These coefficients seem to be analytically derived/derivable, only

the generating scheme was not documentd. However I provided a method

how to produce these coefficients for finitely, but arbitrary many

terms of the series for U_s^(m)(x)) by symbolic exponentiation of

the parametrized matrix-logarithm, so at least that is a *basis* for

the analysis of convergence and summability of the occuring series.

Except for such borderline-cases I would prefer the definition of

the continuous tetration based on the eigenanalysis of the involved

matrix Bs. From the convergent cases -e <log(s) <1/e we have good

numerical evidence, that the eigenvalues are the powerseries of log(h(s)),

where h(s) is Ioannis Galidakis' h-function.

If that hypothesis can be verified, then differentiation with respect

to the height-parameter m means then only analysis with respect to

the fact, that m occurs as power in the exponent of the set of eigenvalues,

and that this is then also infinitely often differentiable.

The structure of the eigenvector-matrices are also not yet analytically known,

although the numerical approximations are already usable, so that the

approximations for real or complex parameters m (for s in the admissible

range) can simply be done based on the real or complex powers of the

set of eigenvalues.

Well, I found already the structure of two eigenvectors - they are

solutions of

[ edited 2.time ]

V(x)~ * Bs = V(x)~ * 1 // where V(x) is of the type [1,x,x^2,x^3,...]

A(x)~ * Bs = A(x)~ * eigenvalue_2 // where A(x) is of the type [0,1*x,2*x^2,3*x^3,...]

A(x) may be written as A(x) = x * V(x)' = x * dV(x)/dx so

V(x)'~ * Bs = V(x)'~ (w.r. to x)

X ~ * Bs = X ~ * eigenvalue_k // structure of X unknown yet

and as such the eigenvector to eigenvalue_1=1 seems simply to reflect

the concept of "fixpoints". (fixpoint for the eigenvalue_k=1).

For s= sqrt(2) we have interestingly 2 possible eigenvectors for the same eigenvalue 1

(which actually occurs only once)

V(2)~ * Bs = V(2)~ * 1

V(4)~ * Bs = V(4)~ * 1

so this is another point for further discussion...

The eigenvector for the second eigenvalue has a simple structure,

but for higher indexes I've not been successful yet.

So far my opinions/thoughts

Gottfried

Hmm, my thoughts may be amateurish, but I've time for a bit of discussion of it currently.

That, what I called "my matrix method" is nothing else than a concise notation for the manipulations of exponential-

series, which constitute the iterates of x, s^x, s^s^x .

The integer-iterates.

It comes out,that the coefficients for the series, which has to be summed to get the value for

have a complicated structure, but which is anyway iterative with a simple scheme.

It is a multisum, difficult to write, so I indicate it for the first three iterates m=1,m=2,m=3

m=1:

m=2:

m=3:

In m>2 the binomial-term expands to a multinomial-term.

When denoting

then the notation for the general case m>1 can be shortened:

m=arbitrary, integer>0:

(hope I didn't make an index-error).

This formula can also be found in wrong reference removed (update)

Convergence:

Because we know, that powertowers of finite height are computable by

exponential-series we know also, that for the finite case this sum-

expression converges.

This is due the weight, that the factorials in the denominators accumulate.

Bound for base-parameter s:

However, in the case of infinite height we already know,

that log(s) is limited to -e < log(s) < 1/e

Differentability:

In the general formula above we find the parameter for s as n'th power

of its logarithm (divided by the n'th factorial). This simply provides

infinite differentiablity with respect to s

Differentiation with respect to the height-parameter m could be described

when the eigensystem-decomposition is used (see below)

(but may be I misunderstood the ongoing discussion about this topic

completely - I kept myself off this discussion due to lack

of experience of matrix-differentiation :-( )

In all, I think, that this description is undoubtedly the unique definition

for the tetration of integer heights in terms of powerseries in s and x.

------------------------------------------------

For the continuous case my idea was, that it may be determined

either by eigenvalue-decomposition of the involved matrices, or,

where this is impossible due to inadmissible parameters s,

by the matrix-logarithm.

The eigenvalue-decomposition has the diasadvantage, that its

properties are not yet analytically known, while matrix-logarithm

is sometimes an option, if the eigenvalue-decomposition

fails since it seems, that the resulting coefficients could be

easier be determined.

An example is the U-iteration U_s^(m)(x) = s^U_s^(m-1)(x) - 1,

where s=e, which we discuss here as "exp(s)-1" - version. For this

case Daniel already provided an array of coefficients for a series

which allows to determine the U-tetration U_s^(m)(x) for fractional,

real and even complex parameters m.

These coefficients seem to be analytically derived/derivable, only

the generating scheme was not documentd. However I provided a method

how to produce these coefficients for finitely, but arbitrary many

terms of the series for U_s^(m)(x)) by symbolic exponentiation of

the parametrized matrix-logarithm, so at least that is a *basis* for

the analysis of convergence and summability of the occuring series.

Except for such borderline-cases I would prefer the definition of

the continuous tetration based on the eigenanalysis of the involved

matrix Bs. From the convergent cases -e <log(s) <1/e we have good

numerical evidence, that the eigenvalues are the powerseries of log(h(s)),

where h(s) is Ioannis Galidakis' h-function.

If that hypothesis can be verified, then differentiation with respect

to the height-parameter m means then only analysis with respect to

the fact, that m occurs as power in the exponent of the set of eigenvalues,

and that this is then also infinitely often differentiable.

The structure of the eigenvector-matrices are also not yet analytically known,

although the numerical approximations are already usable, so that the

approximations for real or complex parameters m (for s in the admissible

range) can simply be done based on the real or complex powers of the

set of eigenvalues.

Well, I found already the structure of two eigenvectors - they are

solutions of

[ edited 2.time ]

V(x)~ * Bs = V(x)~ * 1 // where V(x) is of the type [1,x,x^2,x^3,...]

A(x)~ * Bs = A(x)~ * eigenvalue_2 // where A(x) is of the type [0,1*x,2*x^2,3*x^3,...]

A(x) may be written as A(x) = x * V(x)' = x * dV(x)/dx so

V(x)'~ * Bs = V(x)'~ (w.r. to x)

X ~ * Bs = X ~ * eigenvalue_k // structure of X unknown yet

and as such the eigenvector to eigenvalue_1=1 seems simply to reflect

the concept of "fixpoints". (fixpoint for the eigenvalue_k=1).

For s= sqrt(2) we have interestingly 2 possible eigenvectors for the same eigenvalue 1

(which actually occurs only once)

V(2)~ * Bs = V(2)~ * 1

V(4)~ * Bs = V(4)~ * 1

so this is another point for further discussion...

The eigenvector for the second eigenvalue has a simple structure,

but for higher indexes I've not been successful yet.

So far my opinions/thoughts

Gottfried

Gottfried Helms, Kassel