Tavmjong at 18:07, 15 June 2020

2020-06-15T18:07:42Z

← Older revision		Revision as of 18:07, 15 June 2020
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	The reason for \eqref{eq:W3k} was long a mystery, but it will be explained		The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
	at the end of the paper.		at the end of the paper.

			$$x^{x^{x^{x^{x^{x^{x^{x^{x^{x^{x}}}}}}}}}}$$

Doctormo: Created page with " $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)..."

2020-06-13T12:04:19Z

Created page with " $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)..."

New page

$$\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$$

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.

Scratch - Revision history

Tavmjong at 18:07, 15 June 2020

Doctormo: Created page with " $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)..."