Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials

Consequences of the Darboux transformations for the finite-difference Schrodinger equations and three-term recurrence relations for orthogonal polyno- mials are considered. An equivalence of the chain of these transformations, or discrete dressing chain, to the discrete-time Toda lattice is established. A more fundamental discrete-time Volterra lattice consisting of one simple difference- difference nonlinear equation is found. Some simple similarity reductions of these lattices are described. A group-theoretical meaning of the Darboux transforma- tions is illustrated on the example of Meixner polynomials. The general set of Askey-Wilson potentials is shown to define a class of solutions of the derived discrete-time equations. A subset of the latter potentials that can be obtained by dressing of the free discrete Schrodinger equation is characterized. A class of q-Racah polynomials for qN = 1 orthogonal with respect to a positive measure is obtained by undressing of the finite-dimensional Chebyshev polynomials.

[1]  F. Gesztesy,et al.  Limits of soliton solutions , 1992 .

[2]  T. Chihara,et al.  ON CO-RECURSIVE ORTHOGONAL POLYNOMIALS , 1957 .

[3]  Ryogo Hirota,et al.  Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation , 1977 .

[4]  D. Masson,et al.  Spectral theory of Jacobi matrices in l 2 ( Z ) and the su (1,1) lie algebra , 1991 .

[5]  On the Toda shock problem , 1993 .

[6]  Spiridonov,et al.  Universal superpositions of coherent states and self-similar potentials. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[7]  C. Thompson,et al.  Integrable mappings and soliton equations , 1988 .

[8]  Richard Askey,et al.  A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or 6 - j Symbols. , 1979 .

[9]  V. B. Uvarov,et al.  Classical Orthogonal Polynomials of a Discrete Variable , 1991 .

[10]  M. Crum.  ASSOCIATED STURM-LIOUVILLE SYSTEMS , 1999, physics/9908019.

[11]  L. Vinet,et al.  Periodic reduction of the factorization chain and the Hahn polynomials , 1994 .

[12]  Decio Levi,et al.  Nonlinear differential difference equations as Backlund transformations , 1981 .

[13]  Mourad E. H. Ismail,et al.  Relation between polynomials orthogonal on the unit circle with respect to different weights , 1992 .

[14]  Satoshi Tsujimoto,et al.  Difference Scheme of Soliton Equations , 1993 .

[15]  R. Askey.  Continuous q-Hermite Polynomials When q > 1 , 1989 .

[16]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[17]  B. Simon,et al.  On the Toda and Kac-van Moerbeke systems , 1993 .

[18]  J. Geronimo.  Scattering theory, orthogonal polynomials, and q -series , 1994 .

[19]  J. L. Burchnall,et al.  Commutative Ordinary Differential Operators , 1923 .

[20]  E. Schrödinger.  A method of determining quantum-mechanical eigenvalues and eigenfunctions , 1940 .

[21]  戸田 盛和.  Theory of nonlinear lattices , 1981 .

[22]  F. Grünbaum,et al.  Differential equations in the spectral parameter , 1986 .

[23]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[24]  L. Vinet,et al.  Quantum Mechanics and Polynomials of a Discrete Variable , 1993 .

[25]  P. Santini,et al.  Integrable symplectic maps , 1991 .

[26]  V. Matveev,et al.  Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations I , 1979 .

[27]  David Galant,et al.  An implementation of Christoffel’s theorem in the theory of orthogonal polynomials , 1971 .

[28]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[29]  M. Ismail,et al.  $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals , 1994 .

[30]  A. Shabat.  The infinite-dimensional dressing dynamical system , 1992 .

[31]  V. Matveev,et al.  Differential-difference evolution equations. II (Darboux transformation for the Toda lattice) , 1979 .

[32]  Spiridonov.  Exactly solvable potentials and quantum algebras. , 1992, Physical review letters.

[33]  Richard Askey,et al.  Recurrence relations, continued fractions, and orthogonal polynomials , 1984 .

[34]  B. Kupershmidt,et al.  Time discretizations of lattice integrable systems , 1992 .

[35]  K. Case.  Orthogonal polynomials from the viewpoint of scattering theory , 1974 .

[36]  I. Krichever.  ALGEBRAIC CURVES AND NON-LINEAR DIFFERENCE EQUATIONS , 1978 .

[37]  W. Miller.  Lie theory and difference equations. II , 1969 .

[38]  D. Masson,et al.  Exceptional $q$-Askey-Wilson polynomials and continued fractions , 1991 .

[39]  D. Chudnovsky.  Topological and algebraic structure of linear problems associated with completely integrable systems , 1983 .

[40]  W. Assche,et al.  Sieved orthogonal polynomials and discrete measures with jumps dense in an interval , 1989 .

[41]  Mizan Rahman,et al.  The associated Askey-Wilson polynomials , 1991 .

[42]  Laura M. Chihara.  Askey-Wilson polynomials, kernel polynomials and association schemes , 1993, Graphs Comb..

[43]  V. B. Uvarov.  The connection between systems of polynomials orthogonal with respect to different distribution functions , 1969 .

[44]  A. Zhedanov.  The Toda chain: Solutions with dynamical symmetry and classical orthogonal polynomials , 1990 .

[45]  Laura M. Chihara,et al.  A class of nonsymmetric orthogonal polynomials , 1987 .

[46]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[47]  A. Zhedanov,et al.  Orthogonal polynomials in the lie algebras , 1986 .

[48]  P. Deift.  Applications of a commutation formula , 1978 .

[49]  R. Askey,et al.  Sieved ultraspherical polynomials , 1984 .

[50]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[51]  L. Vinet,et al.  Difference schrödinger operators with linear and exponential discrete spectra , 1993 .

[52]  Reflectionless potentials and soliton series of the KDV equation , 1992 .

[53]  G. Quispel,et al.  The Lattice Gelfand-Dikii hierarchy , 1991 .

[54]  Walter Gautschi,et al.  A set of orthogonal polynomials induced by a given orthogonal polynomial , 1993 .

[55]  Y. Suris.  Discrete time generalized Toda lattices: Complete integrability and relation with relativistic Toda lattices , 1990 .

[56]  J. Geronimo,et al.  ORTHOGONAL POLYNOMIALS ON SEVERAL INTERVALS VIA A POLYNOMIAL MAPPING , 1988 .

[57]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[58]  S. Suslov.  Classical orthogonal polynomials of a discrete variable continuous orthogonality relation , 1987 .

[59]  K. M. Case,et al.  SCATTERING THEORY AND POLYNOMIALS ORTHOGONAL ON THE REAL LINE , 1980 .

[60]  Kiyoshi Sogo,et al.  Toda molecule equation and quotient-difference method , 1993 .

[61]  M. Ismail,et al.  Sieved orthogonal polynomials. VII. Generalized polynomial mappings , 1993 .

[62]  Sergei K. Suslov,et al.  Classical orthogonal polynomials of a discrete variable on nonuniform lattices , 1986 .

[63]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[64]  M. Wadati,et al.  Exactly Solvable Models in Statistical Mechanics , 1988 .

[65]  T. E. Hull,et al.  The factorization method , 1951 .

[66]  D. Levi,et al.  Continuous symmetries of discrete equations , 1991 .

[67]  Jürgen Moser,et al.  On a class of polynomials connected with the Korteweg-deVries equation , 1978 .

[68]  F. Gesztesy,et al.  (m)KdV solitons on the background of quasi-periodic finite-gap solutions , 1996 .

[69]  V.Spiridonov,et al.  Self-Similar Potentials and the q-Oscillator Algebra at Roots of Unity , 1993, hep-th/9304107.