Though the question was not to me, the matrix form of

or by composition is

where E is the Carleman matrix of exp and so on. is the upper triangular pascal matrix. But this does not help here/gives no way to solve the equation. If we however reduce F to its first column on the left side, we can reduce to the first column on the right side . Then and the fate is on its way and .

For the truncated matrices I think for no is because the Jordan normal form of is a Diagonal matrix with pairwise different eigenvalues while the Jordan normal form of has only the eigenvalue 1.

or by composition is

where E is the Carleman matrix of exp and so on. is the upper triangular pascal matrix. But this does not help here/gives no way to solve the equation. If we however reduce F to its first column on the left side, we can reduce to the first column on the right side . Then and the fate is on its way and .

For the truncated matrices I think for no is because the Jordan normal form of is a Diagonal matrix with pairwise different eigenvalues while the Jordan normal form of has only the eigenvalue 1.