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The Ramanujan Journal

, 26:109 | Cite as

Some arithmetic properties of short random walk integrals

  • Jonathan M. Borwein
  • Dirk Nuyens
  • Armin Straub
  • James Wan
Article

Abstract

We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequence of a more general evaluation, a closed form is obtained for the average distance traveled in three steps. This evaluation, as well as its proof, rely on explicit combinatorial properties, such as recurrence equations of the even moments (which are lifted to functional equations). The corresponding general combinatorial and analytic features are collected and made explicit in the case of 3 and 4 steps. Explicit hypergeometric expressions are given for the moments of a 3-step and 4-step walk and a general conjecture for even length walks is made.

Keywords

Random walks Hypergeometric functions High-dimensional integration Analytic continuation 

Mathematics Subject Classification (2000)

60G50 33C20 05A10 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jonathan M. Borwein
    • 1
  • Dirk Nuyens
    • 2
  • Armin Straub
    • 3
  • James Wan
    • 1
  1. 1.CARMAUniversity of NewcastleCallaghanAustralia
  2. 2.K.U.LeuvenLeuvenBelgium
  3. 3.Tulane UniversityNew OrleansUSA

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