ORTHOGONAL POLYNOMIAL APPROACH TO DISCRETE LAX PAIRS FOR INITIAL BOUNDARY-VALUE-PROBLEMS OF THE QD ALGORITHM

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials correspondi ng to a given linear form. Shifts on the linear form give rise to adja cent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover th e discrete-time Toda chain equations of Hirota and of Suris. This appr oach allows us to derive a Backlund transform that relates these two d ifferent discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity c onfinement property is discussed as well.