On Borwein’s Conjectures for Planar Uniform Random Walks

Abstract

Let p n (x) = ∞ 0 J 0 (xt)[J 0 (t)] n xt d t be Kluyver's probability density for n-step uniform random walks in the Euclidean plane. Through connection to a similar problem in 2-dimensional quantum field theory, we evaluate the third-order derivative p ′′′ 5 (0 + ) in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that p n (x), 0 ≤ x ≤ 1 admits a uniformly convergent Maclaurin expansion for all odd integers n ≥ 5, thus settling another conjecture of Borwein.

Wrońskian factorizations and Broadhurst–Mellit determinant formulae

“…The resemblance between (4.29) and (4.30) is not accidental. We refer our readers to [37] for the connection between Kluyver's probability density function and two-scale Bessel moments.…”

Short Walk Adventures

“…serves as a principal constructor of modular forms and functions. No similar formulae are known for W ′ N (0) when N ≥ 7, though the story continues at a different levelsee [14,30,31] for details.…”

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

“…See especially Bloch, Kerr and Vanhove (2015), Samart (2016) and Zhou (2019a). This context offers another instance of the observation of Rogers, Wan and Zucker (2015) of an L-series of an odd-weighted modular form having critical values that are products of gamma functions.…”

“…General approaches to the small-N problem are Borwein, Nuyens, Straub and Wan (2011), Borwein, Straub, Wan and Zudilin (2012), Borwein, Straub and Vignat (2016) and Joyce (2017). Related papers, including some of a more technical nature, are Borwein, Straub and Wan (2013), , Borwein and Sinnamon (2016) and Zhou (2019a). Borwein (2016) offers an introduction and perspective on recent work.…”