Padé approximants on Riemann surfaces and KP tau functions

Abstract The paper has two relatively distinct but connected goals; the first is to define notion of Padé approximation Weyl–Stiltjes transforms on an arbitrary compact Riemann surface higher genus. data consists a contour in and measure it, together with additional datum local coordinate near point divisor degree g . denominators resulting Padé-like also satisfy orthogonality relation are sections appropriate line bundles. A Riemann–Hilbert problem for square matrix rank shown characterize these orthogonal sections, similar fashion ordinary polynomial case. second part extends this idea explore its connection integrable systems. same can be used pairing between sequences locus deformation space where becomes degenerate fixed coincides zeros “tau” function. We show how tau function satisfies Kadomtsev–Petviashvili hierarchy respect either parameters, certain modification 2-Toda when considering whole sequence functions. construction related Krichever algebro-geometric solutions.