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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. | ||
| − | + | [[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.