# Difference between revisions of "Mathtest"

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\begin{align} | \begin{align} | ||

\label{eq:W3k} | \label{eq:W3k} | ||

− | W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. | + | W_3(k) &= \Re \, \pFq32{ \\ \frac12, -\frac k2, -\frac k2}{1, 1}{4}. |

\end{align} | \end{align} | ||

Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. | Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. |

## Revision as of 13:12, 19 March 2013

$

\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.