Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulæ. For all these classes of polynomials a 2× 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Töplitz determinants of the moments of the pseudo-measure. In particular the results imply that any (shifted) Töplitz (Hänkel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.