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Painlevé equations for semiclassical recurrence coefficients: Research problems 96-2

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References

  1. A. I. Aptekarev, A. Branquinho, F. Marcellán (1996):Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation. Pré-Publicaçoes Univ. Coimbra, Dep. Matemática, 96-04.

  2. S. Belmehdi, A. Ronveaux (1994):About nonlinear systems satisfied by the recurrence coefficients of semiclassical orthogonal polynomials. J. Approx. Theory76:351–368.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. V. Chudnovsky (1980):Riemann monodromy problem, isomonodromy deformation equations and completely integrable systems. In Bifurcation Phenomena in Mathematical Physics and Related Topics, Proceedings, Cargèse, 1979 (C. Bardos, D. Bessis, eds.). NATO ASI series C, vol. 54. Dordrecht: Reidel, pp. 385–447.

    Google Scholar 

  4. G. V. Chudnovsky:Padé approximation and the Riemann monodromy problem.Ibidem.

    Google Scholar 

  5. D. V. Chudnovsky, G. V. Chudnovsky (1994):Explicit continued fractions and quantum gravity. Acta Appl. Math.,36:167–185.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. S. Fokas, A. R. Its, A. V. Kitaev (1991)Discrete Painlevé equations and their appearanc in quantum gravity. Comm. Math. Phys.,142:313–344; (1992):The isomonodromy approach to matrix models in 2D quantum gravity. Ibidem,147:395–430.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Laguerre (1885):Sur la réduction en fractions continues d'une fraction qui satisfait à une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels. J. Math. Pures Appl. (4),1:135–165=pp. 685–711 in Oeuvres, vol. II. New York: Chelsea, 1972.

    MATH  Google Scholar 

  8. A. P. Magnus (1995):Painlevé-type differential equations for the recurrence coefficients of semiclassical orthogonal polynomials. J. Comput. Appl. Math.,57:215–317. Preliminary preprint: ftp://unvie6.un.or.at/siam/opsf/pailevemagnus.tex

    Article  MATH  MathSciNet  Google Scholar 

  9. A. P. Magnus (1995):Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud's equations: numerical explorations. Ann. Numer. Math.,2:311–325. Preprint in directory ftp://unvie6.un.or.at/siam/opsf/magnus/

    MATH  MathSciNet  Google Scholar 

  10. A. P. Magnus (preprint):Problem: Painlevé equations for semiclassical recurrence coefficients. ftp://unvie6.un.or.at/siam/opsf/magnus-painleve-problem.tex

  11. J. Nuttall (1984):Asymptotics of diagonal Hermite-Padé polynomials. J. Approx. Theory,42:299–386.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. A. Shohat (1939):A differential equation for orthogonal polynomials. Duke Math. J.,5:401–417.

    Article  MATH  MathSciNet  Google Scholar 

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Magnus, A.P. Painlevé equations for semiclassical recurrence coefficients: Research problems 96-2. Constr. Approx 12, 303–306 (1996). https://doi.org/10.1007/BF02433045

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