Elliptic Integrable Systems Padé Interpolation Table and Biorthogonal Rational Functions

We study recurrence relations and biorthogonality properties for polynomials and rational functions in the problem of the Padé interpolation in the usual scheme and in the scheme with prescribed poles and zeros. The main result is deriving explicit orthogonality and biorthogonality relations for polynomials and rational functions in both schemes. We show that the simplest linear restrictions in the Padé table (so-called, diagonal, anti-diagonal and vertical strings) lead to different explicit types of biorthogonality relations. Finally, we apply our general theory to a concrete example of the Padé interpolation with prescribed poles and zeros on the elliptic grid. This leads to two types of biorthogonality for elliptic hypergeometric functions 12E11. The first type arises from the Kronecker (anti-diagonal) string and coincides with previously known elliptic BRF. The second type arises from the vertical string. It generates a biorthogonality relation in an infinite set of orthogonality points. This biorthogonality relation is assumed to be new.