Grabner, Peter J.; Steinsky, Bertran Asymptotic behaviour of the poles of a special generating function for acyclic digraphs. (English) Zbl 1081.05011 Aequationes Math. 70, No. 3, 268-278 (2005). 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 \(\varepsilon > 0\). Cited in 4 Documents 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 Keywords:Functional differential equation; special generating function; labelled directed acyclic graph; zero; asymptotic behaviour; saddle point method PDFBibTeX XMLCite \textit{P. J. Grabner} and \textit{B. Steinsky}, Aequationes Math. 70, No. 3, 268--278 (2005; Zbl 1081.05011) Full Text: DOI