Asymptotic Distributions of Zeros of Quadratic Hermite–Pade Polynomials Associated with the Exponential Function

The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e} polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$. The polynomials are defined by the relation $$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$ and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials $p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials $p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in 17].