Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve

Starting with a skew-symmetric inner product over an elliptic curve, we propose the concept of elliptic skew-orthogonal polynomials. Inspired by the Landau-Lifshitz hierarchy and its corresponding time evolutions, we obtain the recurrence relation and the -function representation for such a novel class of skew-orthogonal polynomials. Furthermore, a bilinear integral identity is derived through the so-called Cauchy-Stieljes transformation, from which we successfully establish the connection between the elliptic skew-orthogonal polynomials and an elliptic extension of the Pfaff lattice hierarchy.