The bilateral birth-death chain generated by the associated Jacobi polynomials

2023/01/18 by Manuel D. de la Iglesia, de la Iglesia, Manuel D., Claudia Juarez +1
Chemistry · Physics and Astronomy · #33C45 #42C05 #60J10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Molecular spectroscopy and chirality #Probability (math.PR) #Quantum Mechanics and Non-Hermitian Physics #Quantum optics and atomic interactions

paper · pdf · doi:10.48550/arxiv.2301.07582

openalex publication_date 2023/01/18 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28

Abstract

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix which can be interpreted as the one-step transition probability matrix of a discrete-time bilateral birth-death chain with state space on ℤ. We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

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