Asymptotics for quadratic Hermite-Padé polynomials associated with the exponential function
The (diagonal quadratic) Hermite-Padé polynomials \(p_n, q_n, r_n \in\mathcal P_n\) of type I associated with the exponential function \(e^z\) are defined by the relation \[ e_n(z):=p_n(z)+q_n(z)e^z+r_n(z)e^{2z} = O(z^{3n+2})\quad\text{as}\quad z \to 0, \] and the (diagonal quadratic) Hermite-Padé polynomials type of II of degree \(2n\) \({\mathfrak p}_{2n}, {\mathfrak q}_{2n}, {\mathfrak r}_{2n}\in\mathcal P_{2n}\) are defined by the two relations \[ {\mathfrak e}_{1,2n}(z):={\mathfrak p}_{2n}(z)e^z-{\mathfrak q}_{2n}(z)=O(z^{3n+1}) \quad\text{as}\quad z \to 0, \] \[ {\mathfrak e}_{2,2n}(z):={\mathfrak p}_{2n}(z)e^{2z}-{\mathfrak r}_{2n}(z)=O(z^{3n+1}) \quad\text{as}\quad z \to 0. \] The interesting regularity in distributions of zeros of these polynomials, displayed in the paper for \(n=30\) (stars, boxes, diamonds), suggests that there should exist analytic expressions that allow to describe the asymptotic behavior of the zeros, and also the asymptotic behavior of the Hermite-Padé polynomials and the error term \(e_n\) themselves. Results in this direction are the main topic of the paper. Analytic descriptions are given for the arcs, on which the contracted zeros of both sets of these polynomials cluster as \(n\to\infty\). Analytic expressions are also given for the density functions of the asymptotic distributions of these zeros. The description is based on an algebraic function of third degree and on a harmonic function defined on the Riemann surface associated with the algebraic function. The existence and basic properties of the asymptotic distributions of the zeros and the arcs on which these distributions live are proved. The asymptotic relations themselves are only conjectured, but numerical calculations are presented, which demonstrate the plausibility of these conjectures.
diagonal quadratic Hermite-Padé polynomials of type I and type II
Hermite-Padé simultaneous rational approximants
scientific article; zbMATH DE number 1893819