Three friendly walkers

Iwan Jensen

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More than 15 years ago Guttmann and Vöge (2002 J. Stat. Plan. Inference 101 107), introduced a model of friendly walkers. Since then it has remained unsolved. In this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions. The exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a D-finite function. However, ratios of D-finite functions are inherently not D-finite and in this case we prove that the friendly walkers generating function is the solution to a non-linear differential equation with polynomial coefficients, it is in other words D-algebraic. We find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a D-finite function and the generating function for vicious walkers. We obtain an expression for this D-finite function in terms of a hypergeometric function with a rational pullback and its first and second derivatives.

Original languageEnglish
Article number024003
Number of pages14
JournalJournal of Physics. A. Mathematical and Theoretical
Issue number2
Publication statusPublished - 13 Jan 2017


  • asymptotic series analysis
  • D-finite and D-algebraic functions
  • directed walk models
  • exactly solvable models
  • power-series expansions


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