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2010

Cauchy biorthogonal polynomials

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Cauchy biorthogonal polynomials
The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite–Pad´e 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–Pad´e 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...
M. Bertola, M. Gekhtman, J. Szmigielski
Added 28 Jan 2011
Updated 28 Jan 2011
Type Journal
Year 2010
Where JAT
Authors M. Bertola, M. Gekhtman, J. Szmigielski
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