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Abstract
We establish a new connection between moments of n× n random matrices X_{n} and hypergeometric orthogonal polynomials. Specifically, we consider moments ETrXns as a function of the complex variable s∈ C , whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. (J Math Phys 57:111901, 2016) on the timedelay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→ ∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials.
Original language  English 

Pages (fromto)  10911145 
Number of pages  55 
Journal  Communications in Mathematical Physics 
Volume  369 
Issue number  3 
Early online date  6 Feb 2019 
DOIs  
Publication status  Published  1 Aug 2019 
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Professor Francesco Mezzadri
 Probability, Analysis and Dynamics
 School of Mathematics  Professor of Mathematical Physics
 Applied Mathematics
 Mathematical Physics
Person: Academic , Member