Difference between revisions of "Main Page"
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which occurs in the theory of uniform random walk integrals in the plane, | 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, | where at each step a unit-step is taken in a random direction. As such, | ||
the integral \ | the integral \ref{def:Wns} expresses the $s$-th moment of the distance | ||
to the origin after $n$ steps. | 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 \ref{eq:W3k} was long a mystery, but it will be explained | |||
at the end of the paper. | |||
== First Section == | == First Section == | ||
Revision as of 01:17, 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}
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 \, \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 \ref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
First Section
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