Eigenvalues of the Carleman matrix of b^x

[Gottfried]

(...)
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

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

I've applied the eigensystem-solution. This gives as possible solutions x_k for third row:

Code:

row 2 (third row of W, invariant under transformation by Bb, except scaling by eigenvalue u^2
-------------------------------------------
                0
              u*t
      (3*u-2)*t^2
      (6*u-6)*t^3
    (10*u-12)*t^4
    (15*u-20)*t^5
    (21*u-30)*t^6
    (28*u-42)*t^7
    (36*u-56)*t^8
    (45*u-72)*t^9
   (55*u-90)*t^10
  (66*u-110)*t^11
    ...
all with denominator (u-2)

Begin of decoding:
x0  x1  x2  x3  x4  x5  x6   x7  
--------------------------------------
(0   1   3   6  10  15  21  28  ...) *u
-(0   0   1   3   6  10  15  21  ...) *2  
--------------------------------------
  / (u-2)
=====================================================================

This should be a solution for the fourth row; the vector of entries is invariant except scaling by eigenvalue u^3:
row 3 (fourth row of W):
-------------------------------------------
                                   0
                       (u^3+6*u^2)*t
             (5*u^3+12*u^2-18*u)*t^2
         (13*u^3+15*u^2-60*u+18)*t^3
        (26*u^3+12*u^2-132*u+72)*t^4
              (45*u^3-240*u+180)*t^5
       (71*u^3-24*u^2-390*u+360)*t^6
      (105*u^3-63*u^2-588*u+630)*t^7
    (148*u^3-120*u^2-840*u+1008)*t^8
   (201*u^3-198*u^2-1152*u+1512)*t^9
  (265*u^3-300*u^2-1530*u+2160)*t^10
  (341*u^3-429*u^2-1980*u+2970)*t^11
    ...
all with denominator  (u^3-3*u^2-6*u+18)) = (u-3)*u^2-6)

Begin of decoding:
x0  x1  x2  x3  x4  x5  x6   x7   x8   x9
--------------------------------------------------------------------
(0   1   4  10  20  35  56   84  120  165 ...          )*1*u^3
(0   0   1   3   6  10  15   21   28   36 ...          )*1*u^3

+(0   2   4  5   4   0 - 8 - 21 - 40 - 66 -100 -143 ...)*3*u^2
-(0   0   3 10  22  40  65   98  140  192  255  330 ...)*6*u
+(0   0   0  1   4  10  20   35   56   84  120  165 ...)*18
---------------------------------------------------------------------
   / (u-3)/(u^2-6)
=====================================================================

The meaning of this is, that if we assume a set of eigenvalues [1,u,u^2,u^3,...], then we can determine such invariant vectors for each row of W (and then check numerically or analytically), thus we can construct this part of diagonalization (diagonal, W)