ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS
2019 ◽
Vol 107
(3)
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pp. 392-411
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Keyword(s):
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.
2004 ◽
Vol 19
(15)
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pp. 2545-2559
Keyword(s):
2010 ◽
Vol 25
(11)
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pp. 2355-2363
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Keyword(s):
1988 ◽
Vol 29
(12)
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pp. 2659-2665
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1973 ◽
Vol 14
(1)
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pp. 44-61
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