Eigenvalues of the Carleman matrix of b^x

[Gottfried]

Re first part:
Perhaps I just dont understand your explanation but for my taste you work too much with infinite matrices. We already have seen that they may not even have unique inverse, nor a unique diagonalization. We can not carry over the rules of finite matrices to infinite matrices. Also the fixed point choice is not unique, there are infinite many complex fixed points.

Well, one more try:


a) (first row r=0 of W of matrix-equation W*Bb = W * dV(u) )
Proposal: for all columns c it holds
\( \hspace{24} \sum_{k=0}^{\infty} (1*t^k) *((log(b)*c)^k/k! )= (1*t^c) * u^0 \)

Proof:
\( \hspace{24} \sum_{k=0}^{\infty} log(b^{t*c})^k/k! \\ \\[12pt]


\hspace{24} = b^{t*c} \\ \\[12pt]


\hspace{24} = (b^t)^c \\ \\[12pt]


\hspace{24} = (1*t^c) * u^0 \)



b) (second row r=1 of W of matrix-equation)
Proposal: for all columns c it holds
\( \hspace{24} \sum_{k=1}^{\infty} (k*t^k)*((log(b)*c)^k/k!) = (c*t^c) * u^1 \)

Proof:
\(
\hspace{24} \sum_{k=1}^{\infty} k*log(b^{t*c})^k/k! \\ \\[12pt]


\hspace{24} = log(b^{t*c})* \sum_{k=1}^{\infty} log(t^c)^{k-1}/(k-1)! \\ \\[12pt]


\hspace{24} = log(t^c)* t^c \\ \\[12pt]


\hspace{24} = u*c * t^c \\ \\[12pt]


\hspace{24} = (c * t^c) *u^1
\)

c) (third and consecutive rows) --- here we had to insert a valid assumtion about the coefficient x_k at x_k*t^k respectively x_c*t^c

Proposal: \( \hspace{24} \sum_{k=1}^{\infty} (x_k*t^k)*(log(b)*c)^k/k! = (x_c*t^c) * u^2 \)

x_c are to be determined

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Re second part:
...
Hm this is not quite \( \frac{1}{1-u} \) so where is the error?

Nice, I'll help to try to locate it, possibly tomorrow.