Difference between revisions of "Main Page"
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\end{equation} | \end{equation} | ||
$\newcommand{\Re}{\mathrm{Re}\,} | |||
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$ | |||
We consider, for various values of $s$, the $n$-dimensional integral | We consider, for various values of $s$, the $n$-dimensional integral | ||
| Line 35: | Line 37: | ||
\begin{align} | \begin{align} | ||
\label{eq:W3k} | \label{eq:W3k} | ||
W_3(k) &= \Re \, \ | W_3(k) &= \Re \, \pFq 32{\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 00:18, 23 October 2016
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Take a look at this beautifully typesetted equation: \begin{equation} p(n) = \mathcal{C}_n\times {}_1\!F_2\left(n+1;\frac{2n+1}{2}+\frac{\Gamma_\sigma}{\Gamma}, n+\frac{\Gamma_\sigma}{\Gamma}; -\frac{P_\sigma} {2\Gamma} \right) \end{equation}
$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$
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 \ref{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 \, \pFq 32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \ref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
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