Zeros of some bi-orthogonal polynomials

Abstract

Ercolani and McLaughlin have recently shown that the zeros of the bi-orthogonal polynomials with the weight $w(x,y)=\exp [-(V_1(x)+V_2(y)+2cxy)/2]$, relevant to a model of two coupled hermitian matrices, are real and simple. We show that their argument applies to the more general case of the weight $(w_1\ast w_2\ast ...\ast w_j)(x,y)$, a convolution of several weights of the same form. This general case is relevant to a model of several hermitian matrices coupled in a chain. Their argument also works for the weight $W(x,y)=e^{-x-y}/(x+y)$, $0\le x,y<\infty$, and for a convolution of several such weights.

tex mehta.tex, 1 file, 9 pages [SPhT-T01/086], submitted to J. Phys. A