论文标题
在标准欧几里得晶格中首次回报的渐近学和非全面回报特征
On the Asymptotics and the Non-Holonomic Character of First Returns in the Standard Euclidean Lattice
论文作者
论文摘要
我们在$ \ Mathbb {r} $中的标准正交晶格中首次返回随机步行的数量，我们证明这些数字不会形成$ p $ - 递归序列。在此过程中，通过使用组合和几何乘法原理以及Legendre多项式的经典理论，以基本的方式获得了封闭步行数量的已知渐近学。通过证明相关的生成功能是$ g $ functions，我们使用Hadamard卷积的形式在各个方面找到它们的奇异性，并给出它们满足$ d \ leq 5 $的ODE，其中一些似乎是新的。我们使用封闭步行数量的Lucas属性来证明相应的生成函数不可逆转，因为$ G $ - 功能，这立即暗示了第一次返回步行的生成功能不是全面的。我们提出了一些关于渐近系数和ODE的形式的猜想。
We give precise asymptotics to the number of first time returning random walks in the standard orthogonal lattice in $\mathbb{R}$ and we prove that these numbers do not form a $P$-recursive sequence. In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. By showing that the relevant generating functions are $G$-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions and give the ODEs that they satisfy for $d\leq 5$, some of which seem to be new. We use the Lucas property of the number of closed walks to prove that the corresponding generating function is not invertible as a $G$-function, which immediately implies that the generating function of the first time returning walks is not holonomic. We propose a few conjectures on the form of the asymptotic coefficients and of the ODEs.
