The Formal Theory ofBirth - and

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 ooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satissed. Th eorie formelle des processus de naissance et de mort, combinatoire des chemins, et fractions continues R esum e : Les travaux classiques de Karlin-McGregor et Jones-Magnus ont etabli une correspondance g en erale entre les processus de naissnace et de mort en temps continu et les fractions continues de type Stieltjes-Jacobi ainsi que les polyn^ omes orthogonaux associ es. Cette correspondance fondamentale est r e examin ee ici a la lumi ere de la relation connue en analyse combinatoire qui lie chemins valu es et fractions continues, Etant donn e que les trajectoires de la cha^ ne de Markov incluse sont des chemins, les transform ees de Laplace d'un grand nombre de caract eristique transitoires peuvent ^ etre obtenues syst ematiquement. Les applications incluent l'analyse des evolutions dans une bande, des travers ees de bande, ainsi que le temps, l'aire, ou le nombre de transitions sous conditions g eom etriques. 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 ooring and ceiling conditions, as well as time, area, or number of transitions while a …