Three friendly walkers

Abstract

More than 15 years ago Guttmann and Vöge [J. Statist. Plann. Inference, {\bf 101}, 107 (2002)], 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, originally introduced by Tsuchiya and Katori [J. Phys. Soc. Japan {\bf 67}, 1655 (1988)], 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 then show via numerically exact calculations that the generating function of the original model can also be expressed as a D-finite function times the reciprocal of 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.