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<math>\sqrt{\sqrt{\sqrt{x}}}</math>
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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
 +
\begin{align}
 +
  \label{def:Wns}
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  W_n (s)
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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
 +
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}
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  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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[[Category:Wiki/en]]

Aktuelle Version vom 28. Juni 2018, 22:32 Uhr

$

 \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.