Lfairhead le 6 avril 2022 à 15:16

2022-04-06T15:16:30Z

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2020-11-14T16:17:52Z

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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.

← Version précédente		Version du 6 avril 2022 à 15:16
Ligne 25 :		Ligne 25 :
	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.
		+
		+
		+	<syntaxhighlight lang="fortran">
		+	PROGRAM Truc
		+	IMPLICIT NONE
		+	DO i = 1, iim
		+	DO j = 1, jjm
		+	.....
		+	END DO
		+	END DO
		+	END PROGRAM
		+	</syntaxhighlight>

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Lfairhead le 6 avril 2022 à 15:16

Lfairhead : Page créée avec « $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \ri... »