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<p>Hypergeometric multiple orthogonal polynomials on the stepline are shown to be related to Markov chains beyond&#xa0;birth-and-death. The corresponding Hessenberg matrix can be normalized to a stochastic matrix. We find twelve uniform Hessenberg matrices of Toeplitz type and their normalizations to stochastic or semi-stochastic matrices. The corresponding weights and the stochastic bidiagonal factorizations for these matrices are provided. Only one of these uniform stochastic cases, which is a recurrent Markov chain, has a uniform stochastic factorization. Chain of Christoffel transformations connecting the stochastic uniform tuples and the semi-stochastic uniform tuples, between them, are presented.</p>

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Uniform multiple orthogonal polynomials and Markov chains

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月5日

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  • Amílcar Branquinho,
  • Juan E. F. Díaz,
  • Ana Foulquié-Moreno,
  • Manuel Mañas

摘要

Hypergeometric multiple orthogonal polynomials on the stepline are shown to be related to Markov chains beyond birth-and-death. The corresponding Hessenberg matrix can be normalized to a stochastic matrix. We find twelve uniform Hessenberg matrices of Toeplitz type and their normalizations to stochastic or semi-stochastic matrices. The corresponding weights and the stochastic bidiagonal factorizations for these matrices are provided. Only one of these uniform stochastic cases, which is a recurrent Markov chain, has a uniform stochastic factorization. Chain of Christoffel transformations connecting the stochastic uniform tuples and the semi-stochastic uniform tuples, between them, are presented.