Random walks with similar transition probabilities

We consider random walks on the nonnegative integers with a possible absorbing state at -1. A random walk ? is called α-similar to a random walk ? if there exist constants C_ij such that for the corresponding n-step transition probabilities P̃ _ij (n)=α^-nC_ijP_ij (n), i,j≥0, hold. We give necessary and sufficient conditions for the α-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.