About several classes of bi-orthogonal polynomials and discrete integrable systems

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka–Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient–quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

[1]  Yi He,et al.  An Extended Multistep Shanks Transformation and Convergence Acceleration Algorithm with Their Convergence and Stability Analysis , 2013, Numerische Mathematik.

[2]  Yusaku Yamamoto,et al.  Integrable discrete hungry systems and their related matrix eigenvalues , 2013 .

[3]  Claude Brezinski,et al.  Multistep έ-algorithm, Shanks' transformation, and the Lotka-Volterra system by Hirota's method , 2012, Math. Comput..

[4]  Yi He,et al.  Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation , 2011, SIAM J. Sci. Comput..

[5]  Yusaku Yamamoto,et al.  A Bäcklund transformation between two integrable discrete hungry systems , 2011 .

[6]  F. Nijhoff,et al.  Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm , 2010, 1005.0482.

[7]  S. Tsujimoto,et al.  Integrable discrete time chains for the Frobenius-Stickelberger-Thiele polynomials , 2007 .

[8]  Yoshimasa Nakamura,et al.  An application of the discrete Lotka–Volterra system with variable step-size to singular value computation , 2004 .

[9]  Yoshimasa Nakamura,et al.  On the convergence of a solution of the discrete Lotka-Volterra system , 2002 .

[10]  Yoshimasa Nakamura,et al.  The discrete Lotka-Volterra system computes singular values , 2001 .

[11]  Alexei Zhedanov,et al.  Spectral Transformation Chains and Some New Biorthogonal Rational Functions , 2000 .

[12]  Tetsuji Tokihiro,et al.  Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization , 1999 .

[13]  L. Vinet,et al.  An Integrable Chain and Bi-Orthogonal Polynomials , 1998 .

[14]  A. Zhedanov,et al.  Discrete-time Volterra chain and classical orthogonal polynomials , 1997 .

[15]  P. Moerbeke,et al.  String orthogonal polynomials, string equations, and 2 Toda symmetries , 1997, hep-th/9706182.

[16]  Alfred Ramani,et al.  Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm , 1995 .

[17]  B. Grammaticos,et al.  Integrable lattices and convergence acceleration algorithms , 1993 .

[18]  C. Brezinski.  Biorthogonality and its Applications to Numerical Analysis , 1991 .

[19]  Richard A. Brualdi,et al.  Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley , 1983 .

[20]  D. Chudnovsky,et al.  Laws of composition of Bäcklund transformations and the universal form of completely integrable systems in dimensions two and three. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

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

[22]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[23]  H. Rutishauser.  Der Quotienten-Differenzen-Algorithmus , 1954 .

[24]  P. E. Spicer.  On orthogonal polynomials and related discrete integrable systems , 2006 .

[25]  Masashi Iwasaki,et al.  An application of the discrete Lotka–Volterra system with variable step-size to singular value computation , 2004 .

[26]  Alexei Zhedanov,et al.  Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials , 1994 .

[27]  W. Symes.  The QR algorithm and scattering for the finite nonperiodic Toda Lattice , 1982 .