 Scale function of a birth-death chain

A strictly increasing sequence $s_i$ for which $s(X_t)$ is a martingale along the birth-death chain before hitting its boundary. With upward and downward probabilities $p_i,q_i$, harmonicity means $p_i(s_{i+1}-s_i)=q_i(s_i-s_{i-1})$. This coordinate is useful for hitting probabilities and optimal stopping.