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| − | <!-- some LaTeX macros we want to use: -->
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| − | $
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| − | \newcommand{\Re}{\mathrm{Re}\,}
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| − | \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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| − | $
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| − | We consider, for various values of $s$, the $n$-dimensional integral
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| − | \begin{align}
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| − | \label{def:Wns}
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| − | W_n (s)
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| − | &:=
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| − | \int_{[0, 1]^n}
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| − | \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
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| − | \end{align}
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| − | which occurs in the theory of uniform random walk integrals in the plane,
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| − | where at each step a unit-step is taken in a random direction. As such,
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| − | the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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| − | to the origin after $n$ steps.
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| − | By experimentation and some sketchy arguments we quickly conjectured and
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| − | strongly believed that, for $k$ a nonnegative integer
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| − | \begin{align}
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| − | \label{eq:W3k}
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| − | W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
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| − | \end{align}
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| − | Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.
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| − | The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
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| − | at the end of the paper.
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