关于标准欧几里得格中与随机游走相关的微分算子的不可约性
On the Irreducibility of the Differential Operators Associated to Random Walks in the Standard Euclidean Lattice
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中文总结 AI 辅助
研究标准欧几里得格中与随机游走相关序列的生成函数，利用修正博雷尔变换等，证明\(F_d\)被特定不可约算子消去，确定其结构，还得出序列\((x_n^{(d)})_n\)的递归关系及相关多项式系数，对\(A_d\)也有类似成果。
中文摘要 AI 辅助
对于正整数\(d\geq1\)，考虑由\(A_{n}^{(d)} =\sum_{n_1+\dots+n_d=n} \frac{(2n)!}{(n_1!)^2 (n_2!)^2 \dots (n_d!)^2}\)和\(x_{n}^{(d)} = \frac{A_{n}^{(d)}}{\binom{2n}{n}}\)给出的序列\((A_{n}^{(d)})_n\)和\((x_{n}^{(d)})_n\)。聚焦于其生成函数\(A_d\)和\(F_d\)的分析性质。利用修正的博雷尔变换、代数和组合考虑，证明\(F_d\)被一个\(d - 1\)阶的不可约富克斯型微分算子\(L_{d - 1,F}\)消去，确定\(F_d\)作为全局解析函数的结构，还表明序列\((x_n^{(d)})_n\)满足宽度\(r=\lfloor (d + 1)/2 \rfloor\)的最小递归，计算了相关多项式系数，对函数\(A_d\)也得到类似结果。
英文摘要
For a positive integer $d\geq 1$, we consider the sequences $(A_{n}^{(d)})_n$ and $(x_{n}^{(d)})_n$ given by $$ A_{n}^{(d)} =\sum_{n_1+\dots+n_d=n} \frac{(2n)!}{(n_1!)^2 (n_2!)^2 \dots (n_d!)^2} \quad \text{ and } \quad x_{n}^{(d)} = \frac{A_{n}^{(d)}}{\binom{2n}{n}}. $$ They have rich combinatorial interpretations, but we focus on the analytical properties of their generating functions $A_d$ and $F_d$. We use a modified Borel transform, and algebraic and combinatorial considerations to prove that $F_d$ is annihilated by an irreducible Fuchsian differentiable operator $L_{d-1,F}$ of order $d-1$. We determine the structure of $F_d$ as a global analytic function (analytic continuations from the original disk of definition, branches, finite singularities, and the structure of $F_d$ near the finite singularities). Additionally, we show that the sequence $(x_n^{(d)})_n$ satisfies a minimal recurrence of width $r=\lfloor (d+1)/2 \rfloor$ with polynomial coefficients $$ Q_r(n+r)\,x_{n+r}+\cdots + Q_0(n)\,x_n=0, \; n \ge 0. $$ These polynomials are shown to have very specific symmetries and we compute explicitly $Q_0$, $Q_1$, and $Q_r$. Similar results about the functions $A_d$ are obtained.
