标题： 三个友好的步行者 显示英文标题

标题： Three friendly walkers

摘要： 早在15多年前，Guttmann和Vöge [J. Statist. Plann. Inference,{\bf 101}, 107 (2002)] 引入了一个友好行走者模型。自那时以来，该问题一直未得到解决。在本文中，我们提供了对一个密切相关的模型的精确解，该模型最初由Tsuchiya和Katori [J. Phys. Soc. Japan{\bf 67}, 1655 (1988)] 引入，该模型仅在边界条件上有所不同。精确解用反向的恶作剧行走者生成函数来表示，这是一个D-有限函数。然而，D-有限函数的比值本质上不是D-有限的，在这种情况下，我们证明了友好行走者生成函数是具有多项式系数的非线性微分方程的解，换句话说，它是D-代数的。然后通过数值精确计算，我们表明原始模型的生成函数也可以表示为D-有限函数乘以恶作剧行走者的生成函数的倒数。我们得到了这个D-有限函数的表达式，该表达式涉及一个具有有理拉回的${}_{2}F_{1}$超几何函数及其一阶和二阶导数。 显示英文摘要

摘要： 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.