### Abstract

In a recent paper [J. Theor. Probab. 25 (2012) 950-980] Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative integers by establishing a representation for the Laplace transform $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for {\em any} pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ when either $i$ or $j$ tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular {\em associated polynomials} and {\em Markov's Theorem}.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 13 |

Publication status | Published - Apr 2015 |

### Publication series

Name | Memorandum of the Department of Applied Mathematics |
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No. | 2044 |

ISSN (Print) | 1874-4850 |

### Keywords

- IR-95675
- METIS-312553
- First-hitting time
- EWI-25939
- Orthogonal polynomials
- Associated polynomials
- Markov's Theorem
- MSC-42C05
- MSC-60J80
- Birth-death process

## Cite this

van Doorn, E. A. (2015).

*An orthogonal-polynomial approach to first-hitting times of birth-death processes*. (Memorandum of the Department of Applied Mathematics; No. 2044). Enschede: University of Twente, Department of Applied Mathematics.