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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.

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