An urn model for the Jacobi-Piñeiro polynomials

Grünbaum, F.; de la Iglesia, Manuel

doi:10.1090/proc/15910

An urn model for the Jacobi-Piñeiro polynomials
HTML articles powered by AMS MathViewer

by F. Alberto Grünbaum and Manuel D. de la Iglesia PDF

Proc. Amer. Math. Soc. 150 (2022), 3613-3625 Request permission

Abstract:

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.

References

Walter Van Assche and Els Coussement, Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001), no. 1-2, 317–347. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808581, DOI 10.1016/S0377-0427(00)00503-3
A. I. Aptekarev, A. Branquinho, and W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3887–3914. MR 1990569, DOI 10.1090/S0002-9947-03-03330-0
A. Branquinho, A. Foulquié-Moreno, M. Mañas, C. Álvarez-Fernández, and J. E. Fernández-Díaz, Multiple orthogonal polynomials and random walks, arXiv:2103.13715.
J. Bourgain, F. A. Grünbaum, L. Velázquez, and J. Wilkening, Quantum recurrence of a subspace and operator-valued Schur functions, Comm. Math. Phys. 329 (2014), no. 3, 1031–1067. MR 3212879, DOI 10.1007/s00220-014-1929-9
Holger Dette, Bettina Reuther, W. J. Studden, and M. Zygmunt, Matrix measures and random walks with a block tridiagonal transition matrix, SIAM J. Matrix Anal. Appl. 29 (2006/07), no. 1, 117–142. MR 2288017, DOI 10.1137/050638230
Manuel Domínguez de la Iglesia, Orthogonal polynomials in the spectral analysis of Markov processes—birth-death models and diffusion, Encyclopedia of Mathematics and its Applications, [181], Cambridge University Press, Cambridge, 2022. MR 4311159
P. Ehrenfest and T. Eherenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, 8 (1907), 311–314.
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
Lidia Fernández and Manuel D. de la Iglesia, Quasi-birth-and-death processes and multivariate orthogonal polynomials, J. Math. Anal. Appl. 499 (2021), no. 1, Paper No. 125029, 33. MR 4213685, DOI 10.1016/j.jmaa.2021.125029
F. Alberto Grünbaum, Random walks and orthogonal polynomials: some challenges, Probability, geometry and integrable systems, Math. Sci. Res. Inst. Publ., vol. 55, Cambridge Univ. Press, Cambridge, 2008, pp. 241–260. MR 2407600
F. Alberto Grünbaum, An urn model associated with Jacobi polynomials, Commun. Appl. Math. Comput. Sci. 5 (2010), no. 1, 55–63. MR 2600821, DOI 10.2140/camcos.2010.5.55
F. Alberto Grünbaum and Manuel D. de la Iglesia, Stochastic LU factorizations, Darboux transformations and urn models, J. Appl. Probab. 55 (2018), no. 3, 862–886. MR 3877314, DOI 10.1017/jpr.2018.55
F. Alberto Grünbaum and Manuel D. de la Iglesia, Stochastic Darboux transformations for quasi-birth-and-death processes and urn models, J. Math. Anal. Appl. 478 (2019), no. 2, 634–654. MR 3979127, DOI 10.1016/j.jmaa.2019.05.048
F. A. Grünbaum, I. Pacharoni, and J. Tirao, Two stochastic models of a random walk in the $\textrm {U}(n)$-spherical duals of $\textrm {U}(n+1)$, Ann. Mat. Pura Appl. (4) 192 (2013), no. 3, 447–473. MR 3061108, DOI 10.1007/s10231-011-0232-z
F. A. Grünbaum, L. Velázquez, A. H. Werner, and R. F. Werner, Recurrence for discrete time unitary evolutions, Comm. Math. Phys. 320 (2013), no. 2, 543–569. MR 3053772, DOI 10.1007/s00220-012-1645-2
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling, ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999. MR 1674122, DOI 10.1137/1.9780898719734
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. MR 1130396, DOI 10.1090/mmono/092
L. R. Pineĭro Dias, On simultaneous approximations for some collection of Markov functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1987), 67–70, 103 (Russian). MR 884516