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Densities of short uniform random walks. (English) Zbl 1296.33011

An \(n\)-step random walk on the plane is a random walk starting at the origin and consisting an \(n\) consecutive steps of length 1 each taken into a uniformly random direction.
Let \(p_n(x)\) denote the radial density of the distance travelled in \(n\) steps. There is no known closed form expression for \(p_n\) in general, however it is elementary to show that \[ p_2(x)=\frac{2}{\pi\sqrt{4-x^2}}\quad(0\leq x\leq2). \] The expression for \(p_3\), due to Pearson, is much more complicated: \[ p_3(x)=\operatorname{Re}\left(\frac{\sqrt{x}}{\pi^2}K\left(\sqrt{\frac{(x+1)^3(3-x)}{16x}}\right)\right)\quad(0\leq x\leq 3). \] The present authors find a hypergeometric expression and detailed singularity analysis for \(p_3\) and \(p_4\). One of the main theorems of the paper is the following new hypergeometric representation: \[ p_4(x)=\frac{2\sqrt{16-x^2}}{\pi^2 x}\operatorname{Re}\left(\left.{}_3F_2\left(\begin{matrix}\frac12,\frac12,\frac12\\ \frac56,\frac76\end{matrix}\right|\frac{(16-x^2)^3}{108x^4}\right)\right)\quad(0<x<4). \] Some general considerations for all the \(p_n\)’s are also made: it is shown that they satisfy certain differential equations. Relations to Mahler measures are also discussed.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
60G50 Sums of independent random variables; random walks
34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
44A10 Laplace transform
05A19 Combinatorial identities, bijective combinatorics
11F11 Holomorphic modular forms of integral weight
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