Asymptotics for Quadratic Hermite-padé Polynomials Associated with the Exponential Function
نویسنده
چکیده
The asymptotic behavior of quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I and pn, qn, rn ∈ P2n of type II associated with the exponential function are studied. In the introduction the background of the definition of Hermite-Padé polynomials is reviewed. The quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I are defined by the relation pn(z) + qn(z)e z + rn(z)e 2z = O(z) as z → 0, and the polynomials pn, qn, rn ∈ P2n of type II by the two relations pn(z)e z − qn(z) = O(z ) as z → 0, pn(z)e 2z − rn(z) = O(z ) as z → 0. Analytic descriptions are given for the arcs, on which the contracted zeros of both sets of the polynomials {pn, qn, rn} and {pn, qn, rn} cluster as n → ∞. 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 a harmonic function defined on the Riemann surface, which is 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. Numerical calculations are presented, which demonstrate the plausibility of these conjectures.
منابع مشابه
Asymptotic Distributions of Zeros of Quadratic Hermite-pad E Polynomials Associated with the Exponential Function
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