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ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 107 / Issue 3 / December 2019
- Published online by Cambridge University Press:
- 09 October 2019, pp. 392-411
- Print publication:
- December 2019
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- Article
- Export citation
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Let
$p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for 
$n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-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 
$p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers 
$n\geq 5$, thus settling another conjecture of Borwein.
