The formal theory of birth-and-death processes, lattice path combinatoricsand continued fractions

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their as sociated orthogonal polynomials. This fundamental correspondence is revisit ed here in the light of the basic relation between weighted lattice paths a nd continued fractions otherwise known from combinatorial theory. Given tha t sample paths of the embedded Markov chain of a birth-and-death process ar e lattice paths, Laplace transforms of a number of transient characteristic s can be obtained systematically in terms of a fundamental continued fracti on and its family of convergent polynomials. Applications include the analy sis of evolutions in a strip, upcrossing and downcrossing times under floor ing and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.