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| http://traditio-ru.org/wiki/MathJax_%D0%B4%D0%BB%D1%8F_MediaWiki | | http://ia.wikipedia.org/wiki/Wikipedia:LaTeX_symbols |
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| http://traditio-ru.org/wiki/%D0%A2%D1%80%D0%B0%D0%B4%D0%B8%D1%86%D0%B8%D1%8F:%D0%9F%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D1%8B_%D0%BE%D1%84%D0%BE%D1%80%D0%BC%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB | | http://traditio-ru.org/wiki/Справка:Формулы |
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| http://traditio-ru.org/wiki/%D0%A1%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B0:%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B | | http://traditio-ru.org/wiki/Традиция:Примеры_оформления_формул |
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| | http://traditio-ru.org/wiki/MathJax_для_MediaWiki |
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| http://people.cs.kuleuven.be/~dirk.nuyens/Extension_MathJax/ | | http://people.cs.kuleuven.be/~dirk.nuyens/Extension_MathJax/ |
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| http://www.mediawiki.org/wiki/Extension:MathJax/ru | | http://www.mediawiki.org/wiki/Extension:MathJax/ru |
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| <!-- some LaTeX macros we want to use: -->
| | [[Категория:Справочная информация]] |
| $
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| \newcommand{\Re}{\mathrm{Re}\,}
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| \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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| $
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| We consider, for various values of $s$, the $n$-dimensional integral
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| \begin{align}
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| \label{def:Wns}
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| W_n (s)
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| &:=
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| \int_{[0, 1]^n}
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| \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
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| \end{align}
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| which occurs in the theory of uniform random walk integrals in the plane,
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| where at each step a unit-step is taken in a random direction. As such,
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| the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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| to the origin after $n$ steps.
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| By experimentation and some sketchy arguments we quickly conjectured and
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| strongly believed that, for $k$ a nonnegative integer
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| \begin{align}
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| \label{eq:W3k}
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| W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
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| \end{align}
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| Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.
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| The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
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| at the end of the paper.
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