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Zeros of iterated integrals of polynomials. (English) Zbl 0843.30007

The authors mainly study the zero-distributions of the \(n\)-fold indefinite integral \(I_n (p_n)\) of \(n\)-th degree polynomials \(p_n\) with zero constants of integration. A major part of the paper consists in proving that for \(n \to \infty\) the set of limit points of all zeros of \(\sum^n_{k = 0} {2n + 1 \choose n + 1 + k} z^k\) is the closed curve \(\{z \in \mathbb{C} : |1 + z |^2 = 4 |z |\) and \(|z |\geq 1\}\). This result is combined with a well-known Theorem of Schur and Szegö on the location of zeros of “compositions” of two polynomials. As an application the authors obtain the following result on Legendre polynomials \(P_n\). The set of limit points of all zeros \(\neq 0\) of \(I_n (P_n)\), \(n \to \infty\), is the closed curve \(\{z \in \mathbb{C} : |1 - z^2 |= 2 |z |\) and \(|z |\geq 1\}\). As another main result the authors prove that the zeros \(\neq 0\) of \(\sum^{cn}_{k = n + 1} (nz)^k/k!\), \(c - 1 \in \mathbb{N}\), \(n \to \infty\), converge to a closed curve consisting of pieces of \(\{|z |= {c^{c/c - 1} \over e}\}\) and two “Szegö-curves”.
Reviewer: H.-J.Runckel (Ulm)

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30E15 Asymptotic representations in the complex plane
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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