Cauchy Biorthogonal Polynomials
Abstract
The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous HermitePade' 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 fourterm recurrence relation, have relevant ChristoffelDarboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of HermitePade' 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 DegasperisProcesi 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 RiemannHilbert problem.
 Publication:

arXiv eprints
 Pub Date:
 April 2009
 arXiv:
 arXiv:0904.2602
 Bibcode:
 2009arXiv0904.2602B
 Keywords:

 Mathematical Physics
 EPrint:
 38 pages, partially replaces arXiv:0711.4082