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 $p(i,j)^n$ for any M-state chain and any states i and j.
(i) Compute the eigenvalues $X_1, ... , X_M$ of P by solving the characteristic equation.
(ii) If the eigenvalues are distinct then $p(i,j)^n$ has the form (of a polynomial) for some constants $a_1$, ... , $a_M$ (depending on i and j).
If an eigenvalue Y is repeated (once, say) then the general form includes the term $(an+b) Y^n$.
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:
$a_1 * X_1^n + ... + (a_j + a_{j+1} + ... a_{j+m}) * Y^n + ... + a_M * X_M^n$
Instead, the book seems to suggest that the correct form would be:
$a_1 * X_1^n + ... + (a_j + a_{j+1} * n + ... a_{j+m} * n^m) * Y^n + ... + a_M * X_M^n$
Which is it, and why?