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An Integrable Chain and Bi-Orthogonal Polynomials

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Abstract

An integrable chain connected to the isospectral evolution of the polynomials of type R−I introduced by Ismail and Masson is presented. The equations of motion of this chain generalize the corresponding equations of the relativistic Toda chain introduced by Ruijsenaars. We study simple self-similar solutions to these equations that are obtained through separation of variables. The corresponding polynomials are expressed in terms of the Gauss hypergeometric function. It is shown that these polynomials are stable (up to shifts of the parameters) against Darboux transformations of the generalized chain.

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References

  1. Chihara, T.: An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    Google Scholar 

  2. Hendriksen, E. and van Rossum, H.: Orthogonal Laurent polynomials, Indag. Math. (Ser. A) 48 (1986), 17–36.

    Google Scholar 

  3. Ismail, M. E. H. and Masson, D.: Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), 1–40.

    Google Scholar 

  4. Koekoek, R. and Swarttouw, R. F.: The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 94–05, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1994.

  5. Kharchev, S., Mironov, A. and Zhedanov, A.: Faces of relativistic Toda chain, Internat. J. Modern Phys. A 12 (1997), 2675–2724.

    Google Scholar 

  6. Masson, D.: Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials, in: A. Cuyt (ed.), Nonlinear Numerical methods and Rational Approximations, D. Reidel Dordrecht, 1988, pp. 239–257.

  7. Suris, Yu. B.: A discrete-time relativistic Toda lattice, J. Phys. A: Math. and Gen. 29 (1996), 451–465.

    Google Scholar 

  8. Szegő, G.: Orthogonal Polynomials, 4th edn, Amer. Math. Soc., Providence, 1975.

    Google Scholar 

  9. Toda, M.: Theory of Nonlinear Lattices, 2nd edn, Springer Ser. Solid-State Sci. 20, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  10. Watkins, D. S. and Elsner, L.: Self-similar flows associated with the generalized eigenvalue problem, Linear Algebra Appl. 118 (1989), 107–127.

    Google Scholar 

  11. Zhedanov, A.: Bi-orthogonal rational functions and generalized eigenvalue problem, Preprint, CRM-2539 (1998).

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Authors and Affiliations

  1. Centre de Recherches Mathématiques of the Université de Montréal, CP 6128, succ. Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Luc Vinet

  2. Donetsk Institute for Physics and Technology, Donetsk, 340114, Ukraine

    Alexei Zhedanov

Authors
  1. Luc Vinet
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  2. Alexei Zhedanov
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Vinet, L., Zhedanov, A. An Integrable Chain and Bi-Orthogonal Polynomials. Letters in Mathematical Physics 46, 233–245 (1998). https://doi.org/10.1023/A:1007563402749

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  • DOI: https://doi.org/10.1023/A:1007563402749

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