标题： 有限马尔可夫链与多重正交多项式 显示英文标题

标题： Finite Markov chains and multiple orthogonal polynomials

摘要： 本文研究了随机有限矩阵以及由正交多项式和多重正交多项式族构造的相应有限马尔可夫链所对应的回溯矩阵。 文章探讨了转移矩阵的谱理论，利用了正交多项式和多重正交多项式的性质。 得出了若干特性，包括分类、周期性、递归性、平稳状态、遍历性、预期返回时间、时间反向链以及可逆性。 此外，文章揭示了纯生过程和纯灭过程之间的分解关系。 案例研究集中在超几何正交多项式上，在这里所有计算都可以有效地完成。 特别是在Askey方案中，Hahn（不包括Bessel）下的所有后代，如Hahn、Jacobi、Meixner、Kravchuk、Laguerre、Charlier和Hermite，提供了递归可逆生灭有限马尔可夫链的有趣例子。 此外，文章还考虑了多重正交多项式，包括多重Hahn、Jacobi-Piñeiro、第一类Laguerre和第二类Meixner，以及它们的超几何表示，并推导了相应的递归有限马尔可夫链和时间反向链。 显示英文摘要

摘要： This paper investigates stochastic finite matrices and the corresponding finite Markov chains constructed using recurrence matrices for general families of orthogonal polynomials and multiple orthogonal polynomials. The paper explores the spectral theory of transition matrices, utilizing both orthogonal and multiple orthogonal polynomials. Several properties are derived, including classes, periodicity, recurrence, stationary states, ergodicity, expected recurrence times, time-reversed chains, and reversibility. Furthermore, the paper uncovers factorization in terms of pure birth and pure death processes. The case study focuses on hypergeometric orthogonal polynomials, where all the computations can be carried out effectively. Particularly within the Askey scheme, all descendants under Hahn (excluding Bessel), such as Hahn, Jacobi, Meixner, Kravchuk, Laguerre, Charlier, and Hermite, present interesting examples of recurrent reversible birth and death finite Markov chains. Additionally, the paper considers multiple orthogonal polynomials, including multiple Hahn, Jacobi-Pi\~neiro, Laguerre of the first kind, and Meixner of the second kind, along with their hypergeometric representations and derives the corresponding recurrent finite Markov chains and time-reversed chains.