Quadratic Hermite–Padé polynomials associated with the exponential function |
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Authors: | Herbert Stahl |
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Institution: | TFH-Berlin/FB II, Luxemburger Strasse 10, 13353, Berlin, Germany |
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Abstract: | The asymptotic behavior of quadratic Hermite–Padé polynomials
associated with the exponential function is studied for n→∞. These polynomials are defined by the relation (*) pn(z)+qn(z)ez+rn(z)e2z=O(z3n+2) as z→0, where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper. |
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Keywords: | Quadratic Hermite– Padé polynomials of type I Hermite– Padé polynomials of the exponential function Hermite– Padé approximants |
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