论文标题
$ l_1 $ -norm Ball in $ \ mathbb {z}^d $中的随机步行位置的枚举
Enumeration of Random Walk Positions in $L_1$-norm ball in $\mathbb{Z}^d$
论文作者
论文摘要
在本文中，我们主要关注的是推导通用公式，以计算$ n $ step随机步行的可能位置，以$ \ mathbb {z}^d $在每个步骤中具有单位长度，我们将其表示为$ | p_n^{d} | $。对于我们的结果，我们首先提出了计数公式的复发关系：$ | p_n^{d+1} | = | p_n^d | + 2 \ sum_ {k = 0}^{n-1} | p_k^d | $。接下来，我们建议使用生成函数和Faulhaber的公式推导$ | p_n^{d} | $的明确公式的两种方法。最后，我们在公式的矩阵表示中达到了主要定理。
In this paper, we mainly concerned about deriving the general formula to count the possible positions of $n$ step random walk in $\mathbb{Z}^d$ with unit length in each step, which we denoted as $|P_n^{d}|$. For our results, we firstly propose a recurrence relation of the counting formula: $|P_n^{d+1}| = |P_n^d| + 2\sum_{k=0}^{n-1} |P_k^d|$. Next, we propose two methods in deriving the explicit formula of $|P_n^{d}|$ using generating functions and Faulhaber's formula. Finally, we reached our main theorem in the matrix representation of our formula.
