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<!-- some LaTeX macros we want to use: --> | |||
$ | |||
\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} | |||
\label{def:Wns} | |||
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 \eqref{def:Wns} 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} | |||
\label{eq:W3k} | |||
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. | |||
\end{align} | |||
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} \label{def:Wns} 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 \eqref{def:Wns} 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} \label{eq:W3k} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} 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.