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Pi,i+i(t) = Xit + o(t), Pi,i(t) = 1 (Xi + ,Yi)t + 0(t), Pi,i_i(t) = pit + o(t), as t->O, where Xi, pui are constants which may be thought of as the rates of absorption from state i into states i+1, i-1. As a guide to one's intuition it is useful to think of a material particle which moves from integer to neighboring integer, the path function X(t) being the position of the particle at time t. An elegant description of these processes together with a survey of applications may be found in Feller's book [4, Chapter 17]. Using the above order conditions and the Markoffian nature of the process it is easy to show that the infinite matrix P(t) = (Pij(t)), i, j=O, 1, 2, satisfies the equation (1.1) P'(t) = AP(t), t 0,

The differential equations of birth-and-death processes, and the Stieltjes moment problem