I have a problem understanding a passage in Norris' book. If you have access to it, it's page 8. Here is the relevant text:
More generally, the following method may in principle be used to find a formula for $p^{(n)}(i,j)$ for any $M$-state chain and any states $i$ and $j$.
(i) Compute the eigenvalues $\lambda_1, ... , \lambda_M$ of $P$ by solving the characteristic equation.
(ii) If the eigenvalues are distinct then $p^{(n)}(i,j)$ has the form (of a polynomial) for some constants $a_1$, ... , $a_M$ (depending on i and j).
If an eigenvalue $\lambda$ is repeated (once, say) then the general form includes the term $(an+b) \lambda^n$.
The last bit is where I don't understand. Doing the computations by hand it appears to me that regardless of how many times an eigenvalue is repeated in the diagonal form of $P$, you still end up with a series of coefficients depending exclusively on $i$ and $j$, and a polynomial of the form:
$a_1 * \lambda_1^n + ... + (a_j + a_{j+1} + ... a_{j+m}) * \lambda^n + ... + a_M * \lambda_M^n$
Instead, the book seems to suggest that the correct form would be:
$a_1 * \lambda_1^n + ... + (a_j + a_{j+1} * n + ... a_{j+m} * n^m) * \lambda^n + ... + a_M * \lambda_M^n$
Which is it, and why?