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

Chapter I X Mx, t) = 22 Pu(t)Qi(x) 1=0 or equivalently the vector /(*, 0 = P(t)Q(x).This vector satisfies the equation of(x, t) ---= P'(t)Q(x) = P(l)AQ(x) = -xf(x, t), dt and the initial condition fix, 0) = Q(x).Henceor Mx, t) = e-"Qi(x).Now Pn(t) is thejth Fourier coefficient of/,(x, t) and hence