Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture

Abstract

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl_{r+1} irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight. We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Pi\~neiro multiple orthogonal polynomial and others are given by Wronskian type determinants of Jacobi-Pi\~neiro polynomials. As a byproduct we obtain a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.