The height of watermelons with wall

TL;DR

This paper derives asymptotic results for the height distribution of watermelons with p branches and wall, generalizing previous work on plane trees and watermelons with two branches, using advanced analytic techniques.

Contribution

It introduces new asymptotic formulas for moments and the weak limit of the height distribution of watermelons with wall, extending known results to a more general setting.

Findings

Asymptotic formulas for moments of height distribution

Weak limit of height distribution derived

Reciprocity relation for derivatives of Jacobi's theta functions

Abstract

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice on the average height of planted plane trees, and results by Fulmek and Katori et al. on the expected value, respectively the higher moments, 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 generalises the well known reciprocity law for Jacobi's theta functions.