Electronic Journal of Probability

Moments of discrete orthogonal polynomial ensembles

Philip Cohen, Fabio Deelan Cunden, and Neil O’Connell

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We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments of the randomised ensembles are precisely related to the moments of the corresponding equilibrium measures. We also briefly outline how these results can be interpreted as Cauchy-type identities for certain Schur measures.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 72, 19 pp.

Received: 19 September 2019
Accepted: 27 May 2020
First available in Project Euclid: 1 July 2020

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

random matrices factorial moments Charlier, Meixner and Krawtchouk polynomials hypergeometric functions

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Cohen, Philip; Deelan Cunden, Fabio; O’Connell, Neil. Moments of discrete orthogonal polynomial ensembles. Electron. J. Probab. 25 (2020), paper no. 72, 19 pp. doi:10.1214/20-EJP472. https://projecteuclid.org/euclid.ejp/1593568836

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