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Asymptotic behaviour of the poles of a special generating function for acyclic digraphs. (English) Zbl 1081.05011

Summary: Let \(z_{k}\) be the \(k\)th zero of \(\phi(z) = \sum_{n=0}^\infty \frac{(-z)^n}{n!2^{\binom n2}}\) sorted increasingly by modulus from the origin, for \(k \geq 0\). \(\frac{1}{\phi}\) appears as special generating function for acyclic digraphs and \(\phi\) satisfies the functional differential equation \(\phi'(z) = -\phi(\frac z2)\). It is a conjecture of R. W. Robinson that \(z_{k} = (k +1)2^{k} + o(2^{k})\). We show that there is an integer \(K\) such that \(z_{K+k} = (k+1) 2^{k} + o(\frac{2^k}{(k+1)^{1-\varepsilon}})\) for all \(\varepsilon > 0\).

MSC:

05A16 Asymptotic enumeration
05C20 Directed graphs (digraphs), tournaments
39B32 Functional equations for complex functions
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34K25 Asymptotic theory of functional-differential equations
30E15 Asymptotic representations in the complex plane
32A60 Zero sets of holomorphic functions of several complex variables
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