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The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. (English) Zbl 0966.60069
The connection between birth-and-death processes, continued fractions of the Stieltjes-Jacobi type and orthogonal polynomial systems has been investigated by S. Karlin and J. L. McGregor [Trans. Am. Math. Soc. 85, 489-546 (1957; Zbl 0091.13801)], J. A. Murphy and M. R. O’Donohoe [J. Inst. Math. Appl. 16, 57-71 (1975; Zbl 0314.65057)] and W. B. Jones and W. J. Thron [“Continued fractions: Analytic theory and applications” (1980; Zbl 0445.30003)]. This fundamental correspondence is revisited in this paper in the light of the basic relation between weighted lattice paths and continued fractions. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics are obtained by the authors systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G17 Sample path properties
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