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

The Yablonskii-Vorob'ev polynomials yn(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 (PII). Here we define two-variable polynomials Yn(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 PII, namely the so called {\it alternate discrete} PII, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation (PIII). 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 PII, which recovers Jimbo and Miwa's Lax pair for PII in the continuum limit h?0.