Asymptotic Properties of Zeros of Hypergeometric Polynomials |
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Authors: | Peter L Duren Bertrand J Guillou |
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Institution: | Department of Mathematics, University of Michigan, Ann Arbor, Michigan, f1 |
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Abstract: | In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k
the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |zk(1−z)|=kk/(k+1)k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros. |
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Keywords: | hypergeometric polynomials zeros asymptotics lemniscates |
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