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dd
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<!-- some LaTeX macros we want to use: -->
 +
$
 +
  \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}
 +
\end{align}
 +
which occurs in the theory of uniform random walk integrals in the plane,
 +
where at each step a unit-step is taken in a random direction.  As such,
 +
the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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to the origin after $n$ steps.
 +
 +
By experimentation and some sketchy arguments we quickly conjectured and
 +
strongly believed that, for $k$ a nonnegative integer
 +
\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.
 +
The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained
 +
at the end of the paper.

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$ \newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} $

We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \tag{1} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral (1) expresses the $s$-th moment of the distance to the origin after $n$ steps.

By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align} \tag{2} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, (2) also holds for negative odd integers. The reason for (2) was long a mystery, but it will be explained at the end of the paper.

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