One more note about a thing, which is ringing a bell.
The basic definition of the slog derives from the (impossible) inverse (Bb - I)^-1
This expression - for scalar arguments - is just the formula for the geometric series; and if Bb did not have the eigenvalue 1 we could develop the expression into the geometric series of Bb and from this into the iteration-series of b^^0+b^^1+...+b^^h+... .
Incidentally my first interesting connection with tetration was the other formula for the geometric series; that for the alternating geometric series (Bb + I)^-1 , which can be inverted and gives, for the convergent cases (all eigenvalues in Bb<=1 ) the alternating powertowerseries, about which I wrote my first small discussion (and which I call now "iteration-series" for more generality)
Surprisingly it came out, that much likely it can even be used for divergent cases as crosschecked (as far as possible) with the method of shanks-summation.
Things reminded me strongly of the case of Eta/Zeta-function which I reformulated with the geometric series of P (and in turn with the iteration-series in x+1) using (P + I)^-1 arriving at the Euler-numbers and Eta-function and such. However, I avoided the (P - I)^-1 -version in the beginning because of the same problem on non-invertibility; and after finding a workaround similar to Andrews idea, I could build the/a (P - I)^-1 version to eventually surprisingly arrive at the bernoulli-polynomials and the original Faulhaber/Bernoulli/"Zeta"-matrix for summing of like powers.
Now I have 2 aspects to tinker with:
I would like to understand, why and how here comes the analogy with the geometric series of Bb into the play.
The alternating geometric-series from (Bb + I)^-1 -other than the scalar version - is not twosided symmetric for infinite series of positive and of negative heights. While the scalar version sums to the central value, the matrix-version (and from this derived the alternating iteration-series) needs a certain (yet unknown) functional supplement. What about this problem for the (Bb - I)^-1 version?
Gottfried
The basic definition of the slog derives from the (impossible) inverse (Bb - I)^-1
This expression - for scalar arguments - is just the formula for the geometric series; and if Bb did not have the eigenvalue 1 we could develop the expression into the geometric series of Bb and from this into the iteration-series of b^^0+b^^1+...+b^^h+... .
Incidentally my first interesting connection with tetration was the other formula for the geometric series; that for the alternating geometric series (Bb + I)^-1 , which can be inverted and gives, for the convergent cases (all eigenvalues in Bb<=1 ) the alternating powertowerseries, about which I wrote my first small discussion (and which I call now "iteration-series" for more generality)
Surprisingly it came out, that much likely it can even be used for divergent cases as crosschecked (as far as possible) with the method of shanks-summation.
Things reminded me strongly of the case of Eta/Zeta-function which I reformulated with the geometric series of P (and in turn with the iteration-series in x+1) using (P + I)^-1 arriving at the Euler-numbers and Eta-function and such. However, I avoided the (P - I)^-1 -version in the beginning because of the same problem on non-invertibility; and after finding a workaround similar to Andrews idea, I could build the/a (P - I)^-1 version to eventually surprisingly arrive at the bernoulli-polynomials and the original Faulhaber/Bernoulli/"Zeta"-matrix for summing of like powers.
Now I have 2 aspects to tinker with:
I would like to understand, why and how here comes the analogy with the geometric series of Bb into the play.
The alternating geometric-series from (Bb + I)^-1 -other than the scalar version - is not twosided symmetric for infinite series of positive and of negative heights. While the scalar version sums to the central value, the matrix-version (and from this derived the alternating iteration-series) needs a certain (yet unknown) functional supplement. What about this problem for the (Bb - I)^-1 version?
Gottfried
Gottfried Helms, Kassel