On Borwein's conjectures for planar uniform random walks

Abstract

Let $p_n(x)={\int }_0^{\infty }{J}_0(xt)[J_0(t){]}^{n}xt\mathrm{d}t$ 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 $p_5^{\prime \prime \prime }(0^{+})$ 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 $p_n(x),0\le x\le 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\ge 5$, thus settling another conjecture of Borwein.

(v1) 16 pages, 5 TikZ figures. Proof of Borwein's sum rule for ramble integrals $W_n(s)$. An addendum to arXiv:1706.08308 (v2) 17 pages, with updates in references and Theorem 5.1

References in corpus (6)

• High-precision calculation of the 4-loop contribution to the electron g-2 in QED
• Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
• Wrońskian factorizations and Broadhurst-Mellit determinant formulae
• Wick rotations, Eichler integrals, and multi-loop Feynman diagrams
• Some algebraic and arithmetic properties of Feynman diagrams
• On Laporta's 4-loop sunrise formulae

Cited by in corpus (1)

• $\mathbb{Q}$-linear dependence of certain Bessel moments