Abstract
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways (using particles or holes) of computing the correlation functions of a discrete orthogonal polynomial ensemble.
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REFERENCES
F. J. Dyson, Statistical theory of the energy levels of complex systems I, II, III, J. Math. Phys. 3:140–156, 157–165, 166–175 (1962).
M. Gaudin, Sur la loi limite de l'espacement de valuers propres d'une matrics aleatiore, Nucl. Phys. 25:447–458 (1961).
M. Gaudin and M. L. Mehta, On the density of eigenvalues of a random matrix, Nucl. Phys. 18:420–427 (1960).
M. L. Mehta, Random Matrices, 2nd edn. (Academic Press, New York, 1991).
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc. Japan 60:3298–3322 (1991).
A. Borodin and G. Olshanski, Distributions on partitions, point processes, and the hypergeometric kernel, Commun. Math. Phys. 211:335–358 (2000); math/9904010.
A. Borodin and G. Olshanski, z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and b=2 random matrix ensembles, in Random Matrices and Their Applications, MSRI Publications, Vol. 40, 2001; math/9905189.
A. Borodin and G. Olshanski, Harmonic analysis on the infinite-dimesional unitary group and determinantal point processes, preprint, 2001; math/0109194.
K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. 209: 437–476 (2000); math/9903134.
K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153:259–296 (2001); math/9906120.
K. Johansson, Non-intersecting paths, random tilings and random matrices, preprint, 2000; math/0011250.
C. de Boor and E. B. Saff, Finite sequences of orthogonal polynomials connected by a Jacobi matrix, Linear Algebra Appl. 75:43–55 (1986).
O. Macchi, The coincidence approach to stochastic point processes, Adv. Appl. Prob. 7:83–122 (1975).
D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics (Springer, 1988).
A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13:481–515 (2000); math/9905032.
A. Soshnikov, Determinantal random point fields, Russian Math. Surveys 55:923–975 (2000); math/0002099.
M. Hall, Combinatorial Theory (Blaisdell Pub. Co., Waltham, Mass., 1967).
G. Szegö, Orthogonal Polynomials, AMS Colloquium Publications XXIII (Amer. Math. Soc., N.Y., 1959).
R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, available via ftp://ftp.twi.tudelft.nl/' koekoek.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3: More Special Functions (Gordon and Breach, 1990).
W. N. Bailey, Generalized Hypergeometric Series (Cambridge University Press, London, 1935).
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Borodin, A. Duality of Orthogonal Polynomials on a Finite Set. Journal of Statistical Physics 109, 1109–1120 (2002). https://doi.org/10.1023/A:1020432812090
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DOI: https://doi.org/10.1023/A:1020432812090