Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Abstract For a given polynomial P with simple zeros, and semiclassical weight w , we present construction that yields linear second-order differential equation (ODE), in consequence, an electrostatic model for zeros of . The coefficients this ODE are written terms dual call the partner This is absolutely general can be carried out any on complex plane. An additional assumption quasi-orthogonality respect to allows us give more precise bounds degree partner. In case orthogonal quasi-orthogonal polynomials, recover some known results generalize others. Additionally, Hermite–Padé or multiple polynomials type II, approach system equations, from which derive interpretation their vector equilibrium. More detailed obtained special cases Angelesco, Nikishin, generalized Nikishin systems. We also discuss discrete-to-continuous transition these models asymptotic regime, as number tends infinity, into equilibrium problems. Finally, how ODEs third-order well described literature. finish paper by presenting several illustrative examples.