Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair


When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

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  1. 1.

    Ammar, G.S., Gragg, W.B.: Schur flows for orthogonal Hessenberg matrices. In: Hamiltonian and Gradient Flows, Algorithms and Control. Fields Inst. Commun., vol. 3, pp. 27–34. American Mathematical Society, Providence (1994)

    Google Scholar 

  2. 2.

    Bruschi, M., Ragnisco, O.: Recursion operator and Bäcklund transformations for the Ruijsenaars–Toda lattice. Phys. Lett. A 129, 21–25 (1988)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bruschi, M., Ragnisco, O.: Lax representation and complete integrability for the periodic relativistic Toda lattice. Phys. Lett. A 134, 365–370 (1989)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications Series. Gordon & Breach, New York (1978)

    Google Scholar 

  6. 6.

    Common, A.K.: A solution of the initial value problem for half-infinite integrable lattice systems. Inverse Probl. 8, 393–408 (1992)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Common, A.K., Hafez, S.T.: Linearization of the relativistic and discrete-time Toda lattices for particular boundary conditions. Inverse Probl. 8, 59–69 (1992)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Costa, M.S., Felix, H.M., Sri Ranga, A.: Orthogonal polynomials on the unit circle and chain sequences. J. Approx. Theory 173, 14–32 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Coussement, J., Kuijlaars, A., Van Assche, W.: Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials. Inverse Probl. 18, 923–942 (2002)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Daruis, L., Njåstad, O., Van Assche, W.: Para-orthogonal polynomials in frequency analysis. Rocky Mt. J. Math. 33, 629–645 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York University, Courant Institute of Mathematical Sciences. AMS, Providence (1999)

    Google Scholar 

  12. 12.

    Felix, H.M., Sri Ranga, A., Veronese, D.O.: Kernel polynomials from L-orthogonal polynomials. Appl. Numer. Math. 61, 651–665 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Golinskii, L.: Schur flows and orthogonal polynomials on the unit circle. Sb. Math. 197, 1145–1165 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in one Variable. Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  15. 15.

    Jones, W.B., Njåstad, O., Thron, W.J.: Two point Padé expansions for a family of analytic functions. J. Comput. Appl. Math. 9, 105–123 (1983)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jones, W.B., Thron, W.J., Waadeland, H.: A strong Stieltjes moment problem. Trans. Am. Math. Soc. 206, 503–528 (1980)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kharchev, S., Mironov, A., Zhedanov, A.: Faces of relativistic Toda chain. Int. J. Mod. Phys. A 12, 2675–2724 (1997)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mukaihira, A., Nakamura, Y.: Schur flow for orthogonal polynomials on the unit circle and its integrable discretization. J. Comput. Appl. Math. 139, 75–94 (2002)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Nakamura, Y.: A new approach to numerical algorithms in terms of integrable systems. In: Proceedings of the International Conference on Informatics Research for Development of Knowledge Society Infrastructure, ICKS, pp. 194–205 (2004)

    Google Scholar 

  20. 20.

    Nenciu, I.: Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle. Int. Math. Res. Not. 11, 647–686 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Peherstorfer, F.: On Toda lattices and orthogonal polynomials. J. Comput. Appl. Math. 133, 519–534 (2001)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ruijsenaars, S.N.M.: Relativistic Toda systems. Commun. Math. Phys. 133, 217–247 (1990)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Silva, A.P., Sri Ranga, A.: Polynomials generated by a three term recurrence relation: bounds for complex zeros. Linear Algebra Appl. 397, 299–324 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society Colloquium Publications, vol. 54. AMS, Providence (2005)

    Google Scholar 

  25. 25.

    Sri Ranga, A.: Symmetric orthogonal polynomials and the associated orthogonal L-polynomials. Proc. Am. Math. Soc. 123, 3135–3141 (1995)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Sri Ranga, A.: Companion orthogonal polynomials. J. Comput. Appl. Math. 75, 23–33 (1996)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Sri Ranga, A., Andrade, E.X.L.: Zeros of polynomials which satisfy a certain three term recurrence relation. In: Comm. Anal. Theory Contin. Fractions, vol. 1, pp. 61–65 (1992)

    Google Scholar 

  28. 28.

    Sri Ranga, A., Andrade, E.X.L., McCabe, J.H.: Some consequences of symmetry in strong distributions. J. Math. Anal. Appl. 193, 158–168 (1995)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Sri Ranga, A., McCabe, J.H.: On pairwise related strong Stieltjes distributions. K. Nor. Vidensk. Selsk. 3, 3–12 (1996)

    MATH  Google Scholar 

  30. 30.

    Sri Ranga, A., Van Assche, W.: Blumenthal’s theorem for Laurent orthogonal polynomials. J. Approx. Theory 117, 255–278 (2002)

    MathSciNet  Article  Google Scholar 

  31. 31.

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

    MathSciNet  Article  Google Scholar 

  32. 32.

    Suris, Y.B.: New integrable systems related to the relativistic Toda lattice. J. Phys. A, Math. Gen. 30, 1745–1761 (1997)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Suris, Y.B.: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, vol. 219. Birkhäuser, Basel (2003)

    Google Scholar 

  34. 34.

    Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, vol. 23. AMS, Providence (1975)

    Google Scholar 

  35. 35.

    Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22, 431–436 (1967)

    Article  Google Scholar 

  36. 36.

    Zhedanov, A.: The “classical” Laurent biorthogonal polynomials. J. Comput. Appl. Math. 98, 121–147 (1998)

    MathSciNet  Article  Google Scholar 

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The authors are grateful to the anonymous referees for a careful reading of the manuscript.

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Correspondence to Cleonice F. Bracciali.

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The first and third authors are supported by funds from FAPESP (2016/09906-0, 2017/12324-6) and CNPq (305073/2014-1, 305208/2015-2, 402939/2016-6) of Brazil. The second author was supported by grant from CAPES of Brazil.

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Bracciali, C.F., Silva, J.S. & Sri Ranga, A. Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair. Acta Appl Math 164, 137–154 (2019). https://doi.org/10.1007/s10440-018-00229-x

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  • Relativistic Toda lattice
  • Lax pairs
  • L-orthogonal polynomials
  • Kernel polynomials on the unit circle

Mathematics Subject Classification (2000)

  • 34A33
  • 42C05
  • 33C47
  • 47E05