Abstract: In this paper, the main properties of the dual q-matrix Q^* of the generalized birth-death matrix Q were discussed. According to these properties, the digital descriptions of Q^* is strong zero-entrance and zero-exit were found respectively. Following conclusions were got: Q is strong zero-entrance if and only if Q^* is zero-exit; Q is zero-exit if and only if Q^* is strong zero-entrance. Thus the relation between Q and Q^* were found. Besides, the main properties of the minimal Q^*-function were discussed, and the following conclusion was got, under certain conditions, the dual of the minimal Q-function is the minimal Q^*-function exactly.

Key words: generalized birth-death matrix, dual q-matrix, minimal Q^*-function