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Asymptotic properties of zeros of hypergeometric polynomials. (English) Zbl 0983.33008
The zeros of the hypergeometric polynomials $$F(-n, kn+1; kn+2; z)$$ are considerd for any $$k>0$$. It is shown that every point of a certain lemniscate is a cluster point of zeros as $$n\to\infty$$. In an earlier paper by K. Driver and P. Duren [Numer. Algorithms 21, No. 1-4, 147-156 (1999; Zbl 0935.33004)] a similar study for positive integer values of $$k$$ has been given.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
##### Keywords:
hypergeometric polynomials; zeros; asymptotics
Zbl 0935.33004
Full Text:
##### References:
 [1] Borwein, P.B.; Chen, W., Incomplete rational approximation in the complex plane, Constr. approx., 11, 85-106, (1995) · Zbl 0820.41013 [2] de Bruijn, N.G., Asymptotic methods in analysis, (1958), North-Holland Amsterdam · Zbl 0082.04202 [3] Copson, E.T., Asymptotic expansions, (1965), Cambridge University Press London · Zbl 0123.26001 [4] Driver, K.; Duren, P., Asymptotic zero distribution of hypergeometric polynomials, Numer. algorithms, 21, 147-156, (1999) · Zbl 0935.33004 [5] Marden, M., Geometry of polynomials, (1966), American Mathematical Society Providence · Zbl 0173.02703 [6] Whittaker, E.T.; Watson, G.N., Modern analysis, (1943), Macmillan New York · Zbl 0108.26903
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