Cauchy Biorthogonal Polynomials

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dc.creatorBertola, M.
dc.creatorGekhtman, M.
dc.creatorSzmigielski, J.
dc.date2009-04-16
dc.date.accessioned2026-07-07T13:05:35Z
dc.date.available2026-07-07T13:05:35Z
dc.descriptionThe paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes 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 formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.
dc.description38 pages, partially replaces arXiv:0711.4082
dc.identifierhttps://arxiv.org/abs/0904.2602
dc.identifierhttp://arxiv.org/abs/0904.2602
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/227535
dc.subjectMathematical Physics
dc.titleCauchy Biorthogonal Polynomials
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