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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$$\cdot \cdot \cdot$$\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 $\leq x, y < \infty$, and for a convolution of several such weights.
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Madan Lal Mehta. Zeros of some bi-orthogonal polynomials. Journal of Physics A: Mathematical and General (1975 - 2006), IOP Publishing, 2002, 35 (3), pp.517-525. ⟨10.1088/0305-4470/35/3/305⟩. ⟨cea-02904735⟩

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