Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when $h=0$. They also provide rational solutions for a particular discretisation of $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.Comment: 23 pages, IOP style. Title changed, and connection with Umemura polynomials adde

A solution of the initial value problem for half-infinite integrable lattice systems

In previous studies solutions of a number of half-infinite nonlinear lattice systems were constructed from continued fraction solutions to corresponding Riccati equations. A method for linearizing the Kac-Van Moerbeke lattice equations was reconstructed and extended to the discrete nonlinear Schrodinger equation, relativistic Toda lattice equations as well as other examples. This approach demonstrated the important role played by the boundary condition at the finite end and solutions were obtained for given behaviour of this end with time. Here the initial value problem will be solved, i.e. we will obtain solutions of these half-infinite lattice equations corresponding to prescribed values at t = 0. Such solutions were obtained for the Kac-Van Moerbeke lattice through studying the time behaviour of continued fractions related to Jacobi matrices and the corresponding 'hamburger moment problem'. A similar approach is used here and we find for the discrete nonlinear Schrodinger equation that the continued fractions which arise are related to the 'trigonometric moment problem'. We also consider the discrete modified KdV equation, relativistic Toda lattice and discrete-time Toda lattices and in these cases T-fractions, which are related to the 'strong Stieltjes moment problem', are used to solve the initial value problem

BI-Axial Gegenbauer Functions of the 2nd Kind

Bi-axially symmetric monogenic generating functions on R(p divided by q) have been used recently to define generalisations of Gegenbauer polynomials. These polynomials are orthogonal on the unit ball in R(p). Generalised Cauchy transforms of these polynomials are used to define corresponding bi-axial Gegenbauer functions of the second kind. It is demonstrated that these functions of the second kind satisfy second order differential equations related to those satisfied by the corresponding bi-axial Gegenbauer polynomials. (C) 1995 Academic Press, Inc

Singularity Confinement in the Discrete-Time Toda Lattice and Q-D Algorithm

An integrability criterion for discrete systems based on singularity confinement has been defined recently as an analogue of the Painleve test for continuous systems. It will be demonstrated that the discrete time Toda lattice, equivalent to the Q-D algorithm for finding poles of a holomorphic function, satisfies the criterion

The half-infinite discretized hirota equation and the trigonometric moment problem

The discretized Hirota equation is considered for the half-infinite case. Solutions are constructed, which are related to the trigonometric moment problem, by considering continued fraction solutions to a corresponding Riccati equation. It is demonstrated how the latter may be linearized when a certain boundary condition at the finite end is specified. These solutions may be chosen so that they tend to zero at infinity for all time

The Symplectic Matrix Riccati System and a discrete form of an equation of the Chazy XII classification

An example of a non-linear third order differential equation in the Chazy XII classification is shown to be equivalent to a Symplectic Riccati System. This relationship is then used to obtain a discrete form of the above differential equation and both are linearisable. (C) 1999 Elsevier Science Ltd. All rights reserved

Linearization of the Relativistic and Discrete-Time Toda-Lattices for Particular Boundary-Conditions

In a previous study we considered solutions to the Kac-Van Moerbeke and semi-infinite Toda, discrete modified KdV and nonlinear Schrodinger equations. Using the AKNS approach, solutions of these equations were related to continued-fraction solutions of certain Riccati equations. A method for linearizing the Kac-Van Moerbeke lattice was rederived and extended to all the above lattices. Our approach demonstrated the crucial role played by the boundary condition at the finite end. This study is extended here to the relativistic and discrete-time Toda lattices which have been introduced recently

Special Bi-Axial Monogenic Functions

In this paper we extend our recent work on axial monogenic functions in R(m+1) to functions which are monogenic in bi-axially symmetric domains of R(p+q). We show that an integral transform of a wide class of holomorphic functions of a single complex variable gives monogenic functions of this type. It is demonstrated that these integral transforms are related to plane wave monogenic functions. A bi-axial monogenic exponential function is defined using the exponential function of a complex variable and bounds are obtained on its modulus. Bi-axially symmetric monogenic generating functions are used to define generalisations of Gegenbauer polynomials and Hermite polynomials. Finally, bi-axial power functions are constructed using the above integral transform. (C) 1994 Academic Press, Inc