James Wan

DENSITIES OF SHORT UNIFORM RANDOM WALKS

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

Lattice sums then and now

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung’s constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered. How To Order

Three-Step and Four-Step Random Walk Integrals

We investigate the moments of 3-step and 4-step uniform random walks in the plane. In particular, we further analyze a formula conjectured in [Borwein et al. 11] expressing 4-step moments in terms of 3-step moments. Diverse related results including hypergeometric and elliptic closed forms for W 4(±1) are given, and two new conjectures are recorded.

Log-sine evaluations of Mahler measures, II

Abstract. We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in “Log-sine evaluations of Mahler measures” and “Special values of generalized log-sine integrals” by two of the authors. This motivates a detailed study of various multiple polylogarithms and worked examples are given. Our techniques enable the reduction of several multiple Mahler measures, and supply an easy proof of two conjectures by Boyd.

Random walks, elliptic integrals and related constants

In the first quarter of this dissertation, we investigate the problem of how far a walker travels after n unit steps, each taken along a uniformly random direction; the short-step behaviour of this random walk was unknown. Utilising functional equations, we fully analyse the threeand four-step walks, finding the moments and densities of the distance from the origin. Our methods involve a blend of combinatorics, probability, and complex analysis. The derivatives of random walk moments turn out to be Mahler measures. We fruitfully study them using elementary techniques (different to those used by other researchers), namely generating functions of log-sine integrals and trigonometry. On the other hand, some random walk moments can be written as moments of products of complete elliptic integrals. These are studied, culminating in a complete solution for the moments of the product of two elliptic integrals. We also give some results when more elliptic integrals are involved. These endeavours occupy the second quarter of this dissertation. A spectacular application of elliptic integrals is their ability to produce rational series which converge to 1/π, as observed by Ramanujan. Using modular forms and hypergeometric transforms, we produce new classes of 1/π series which involve Legendre polynomials and Apery-like sequences. We give a diverse range of series for related constants, including some based on Legendre’s relation. The third quarter of this dissertation is devoted to this topic. In the last quarter we apply experimental methods to better understand a number of areas encountered in our prior investigations. We simplify proofs for some multiple zeta value identities, give new ones and outline how they may be found. We give a method to quickly generate contiguous relations for hypergeometric series. Lastly, we look at orthogonal polynomials, in particular a new application of Gaussian quadrature to multi-dimensional lattice sums.

Some notes on weighted sum formulae for double zeta values

We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums involving the harmonic numbers, the alternating double zeta values, and the Mordell–Tornheim double sum. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.