### Abstract

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 language | English |
---|---|

Article number | 024003 |

Number of pages | 14 |

Journal | Journal of Physics. A. Mathematical and Theoretical |

Volume | 50 |

Issue number | 2 |

DOIs | |

Publication status | Published - 13 Jan 2017 |

### Keywords

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