Random walks with similar transition probabilities

We consider random walks on the nonnegative integers with a possible absorbing state at -1. Two such random walks X and Y are called k-similar if there exist constants C(i,j) such that for the n-step transition probabilities $\Pw_{ij}(n)=k^{-n}C(i,j)P_{ij}(n)$ hold. We give necessary and sufficient conditions for the k-similarity of two random walks both in terms of the parameters and in terms of the corresponding spectral measures which appear in the spectral representation of the n-step transition probabilities developed by Karlin and McGregor.