The height of watermelons with wall

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalizes a famous result of de Bruijn et al (1972 Graph Theory and Computing (New York: Academic) pp 15–22) on the average height of planted plane trees, and results by Fulmek (2007 Electron. J. Combin. 14 R64) and Katori et al (2008 J. Stat. Phys. 131 1067–83) on the expected value and higher moments, respectively, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information, we prove a reciprocity relation satisfied by the derivatives of one of Jacobi’s theta functions, which generalizes the well-known reciprocity law for Jacobi’s theta functions.

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