HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation

The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions

Philippe Flajolet 1 Fabrice Guillemin
1 ALGO - Algorithms
Inria Paris-Rocquencourt
Abstract : Classic works of Karlin-McGregor and Jones-Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that trajectories of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained 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.
Document type :
Complete list of metadata

Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Wednesday, May 24, 2006 - 11:34:19 AM
Last modification on : Friday, February 4, 2022 - 3:10:28 AM
Long-term archiving on: : Thursday, March 24, 2011 - 12:24:22 PM


  • HAL Id : inria-00073005, version 1



Philippe Flajolet, Fabrice Guillemin. The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions. [Research Report] RR-3667, INRIA. 1999. ⟨inria-00073005⟩



Record views


Files downloads