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Abstract: The paper investigates the properties of certain biorthogonal polynomialsappearing in a specific simultaneous Hermite-Pade- approximation scheme.Associated to any totally positive kernel and a pair of positive measures onthe positive axis we define biorthogonal polynomials and prove that theirzeroes are simple and positive. We then specialize the kernel to the Cauchykernel 1-{x+y} and show that the ensuing biorthogonal polynomials solve afour-term recurrence relation, have relevant Christoffel-Darboux generalizedformulae, and their zeroes are interlaced. In addition, these polynomial solvea combination of Hermite-Pade- approximation problems to a Nikishin system oforder 2. The motivation arises from two distant areas; on one side, in thestudy of the inverse spectral problem for the peakon solution of theDegasperis-Procesi equation; on the other side, from a random matrix modelinvolving two positive definite random Hermitian matrices. Finally, we show howto characterize these polynomials in term of a Riemann-Hilbert problem.

Author: M. Bertola, M. Gekhtman, J. Szmigielski



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