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Duality of Orthogonal Polynomials on a Finite Set

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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

  1. 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).

    Google Scholar 

  2. M. Gaudin, Sur la loi limite de l'espacement de valuers propres d'une matrics aleatiore, Nucl. Phys. 25:447–458 (1961).

    Google Scholar 

  3. M. Gaudin and M. L. Mehta, On the density of eigenvalues of a random matrix, Nucl. Phys. 18:420–427 (1960).

    Google Scholar 

  4. M. L. Mehta, Random Matrices, 2nd edn. (Academic Press, New York, 1991).

    Google Scholar 

  5. T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc. Japan 60:3298–3322 (1991).

    Google Scholar 

  6. A. Borodin and G. Olshanski, Distributions on partitions, point processes, and the hypergeometric kernel, Commun. Math. Phys. 211:335–358 (2000); math/9904010.

    Google Scholar 

  7. 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.

  8. A. Borodin and G. Olshanski, Harmonic analysis on the infinite-dimesional unitary group and determinantal point processes, preprint, 2001; math/0109194.

  9. K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. 209: 437–476 (2000); math/9903134.

    Google Scholar 

  10. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153:259–296 (2001); math/9906120.

    Google Scholar 

  11. K. Johansson, Non-intersecting paths, random tilings and random matrices, preprint, 2000; math/0011250.

  12. C. de Boor and E. B. Saff, Finite sequences of orthogonal polynomials connected by a Jacobi matrix, Linear Algebra Appl. 75:43–55 (1986).

    Google Scholar 

  13. O. Macchi, The coincidence approach to stochastic point processes, Adv. Appl. Prob. 7:83–122 (1975).

    Google Scholar 

  14. D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics (Springer, 1988).

  15. A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13:481–515 (2000); math/9905032.

    Google Scholar 

  16. A. Soshnikov, Determinantal random point fields, Russian Math. Surveys 55:923–975 (2000); math/0002099.

    Google Scholar 

  17. M. Hall, Combinatorial Theory (Blaisdell Pub. Co., Waltham, Mass., 1967).

    Google Scholar 

  18. G. Szegö, Orthogonal Polynomials, AMS Colloquium Publications XXIII (Amer. Math. Soc., N.Y., 1959).

    Google Scholar 

  19. 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.

  20. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3: More Special Functions (Gordon and Breach, 1990).

  21. W. N. Bailey, Generalized Hypergeometric Series (Cambridge University Press, London, 1935).

    Google Scholar 

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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

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