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# Densities of Short Uniform Random Walks

Published:2011-11-03
Printed: Oct 2012
• Jonathan M. Borwein,
CARMA, University of Newcastle, Australia
• Armin Straub,
Tulane University, New Orleans, LA, USA
• James Wan,
CARMA, University of Newcastle, Australia
• Wadim Zudilin,
CARMA, University of Newcastle, Australia
 Format: LaTeX MathJax PDF

## Abstract

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.
 Keywords: random walks, hypergeometric functions, Mahler measure
 MSC Classifications: 60G50 - Sums of independent random variables; random walks 33C20 - Generalized hypergeometric series, ${}_pF_q$ 34M25 - Formal solutions, transform techniques 44A10 - Laplace transform

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