Cauchy biorthogonal polynomials

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite–Padé approximation scheme. Associated with any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeros are simple and positive. We then specialize the kernel to the Cauchy kernel 1x+y and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel–Darboux generalized formulas, and their zeros are interlaced. In addition, these polynomials solve a combination of Hermite–Padé approximation problems to a Nikishin system of order 22. The motivation arises from two distant areas; on the one hand, in the study of the inverse spectral problem for the peakon solution of the Degasperis–Procesi equation; on the other hand, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in terms of a Riemann–Hilbert problem.