Three friendly walkers

TL;DR

This paper provides an exact solution to a model of friendly walkers, revealing its generating function is D-algebraic and expressing it through hypergeometric functions, advancing understanding of lattice path models.

Contribution

It offers the first exact solution to a closely related friendly walker model, showing its generating function is D-algebraic and connecting it to hypergeometric functions.

Findings

Generated function is D-algebraic, satisfying a nonlinear differential equation.

Expressed the generating function in terms of hypergeometric functions with rational pullback.

Linked the friendly walkers model to vicious walkers via generating function ratios.

Abstract

More than 15 years ago Guttmann and V\"oge [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.