A rational analogue of the Krall polynomials (Q2716911)

The Krall polynomials are orthogonal polynomials that are also eigenfunctions of a differential operator. The authors exhibit an analogue of the Krall polynomials within the context of rank-one commutative rings of difference operators. The corresponding spectral curves are unicursal curves with equations \(\nu^2=u^{2R+1}(u+1)^{2S+1}, R,S = 0,1,2,\ldots\). The authors' analogues of the Krall polynomials are rational functions, which satisfy an orthogonality relation on the circle. The proof of the orthogonality relations combines the discrete Kadomtsev-Petviashvili bilinear identities, the cuspidal character of the singularities of the spectral curves, together with an extra symmetry of the problem.