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 1 x+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 2. 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. c © 2009 Elsevier Inc. All rights reserved.