# Three friendly walkers

• Published in 2016
In the collections
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 ${{}_{2}}{{F}_{1}}$ hypergeometric function with a rational pullback and its first and second derivatives.

## Comment

Contains objects called vicious, friendly and super-friendly watermelons. I have no idea why.

### BibTeX entry

@article{Threefriendlywalkers,
title = {Three friendly walkers},
abstract = {More than 15 years ago Guttmann and V{\"{o}}ge (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 ${\{}{\{}{\}}{\_}{\{}2{\}}{\}}{\{}{\{}F{\}}{\_}{\{}1{\}}{\}}$ hypergeometric function with a rational pullback and its first and second derivatives.},
url = {https://iopscience.iop.org/article/10.1088/1751-8121/50/2/024003},
year = 2016,
author = {Iwan Jensen},
comment = {Contains objects called vicious, friendly and super-friendly watermelons. I have no idea why.},
urldate = {2021-03-22},
collections = {attention-grabbing-titles,food}
}