多元Delta-解析函数具有幂律奇点的传递定理
A transfer theorem for multivariate Delta-analytic functions with a power-law singularity
摘要：
本文给出了Flajolet和Odlyzko转移定理的多元推广。类似于单变量的版本，定理假设$Δ$ -解析性（定义的坐标方式）的功能$A(z_1,\ldots,z_d)$在一个独特的优势奇点$(ρ_1,\ldots,ρ_d) \in (\mathbb C_*)^d$，并允许一个翻译，在逐项的基础上，渐近展开的$A(z_1,\ldots,z_d)$周围$(ρ_1,\ldots,ρ_d)$到相应的渐近展开的泰勒系数$a_{n_1,\ldots,n_d}$。我们对待的情况下，渐近展开的$A(z_1,\ldots,z_d)$只包含幂律型条款，其中的指数$n_1,\ldots,n_d$趋于无穷大的一些多项式拉伸对角极限。$a_{n_1,\ldots,n_d}$的渐近展开是\begin{equation*} I(λ_1,\ldots,λ_d) \cdot n_0^{-Θ} \cdot ρ_1^{-n_1}\cdots ρ_d^{-n_d}, \end{equation*}形式的项之和，其中$(λ_1,\ldots,λ_d) \in (0,\infty)^d$是$(n_1,\ldots,n_d)$的拉伸对角极限的方向向量，参数$n_0$以与$n_1,\ldots,n_d$相似的速度趋向于$\infty$，而$Θ\in \mathbb R$和$I:(0,\infty)^d \to \mathbb C$由$A$的渐近展开确定。
英文摘要：
This paper presents a multivariate generalization of Flajolet and Odlyzko's transfer theorem. Similarly to the univariate version, the theorem assumes $Δ$-analyticity (defined coordinate-wise) of a function $A(z_1,\ldots,z_d)$ at a unique dominant singularity $(ρ_1,\ldots,ρ_d) \in (\mathbb C_*)^d$, and allows one to translate, on a term-by-term basis, an asymptotic expansion of $A(z_1,\ldots,z_d)$ around $(ρ_1,\ldots,ρ_d)$ into a corresponding asymptotic expansion of its Taylor coefficients $a_{n_1,\ldots,n_d}$. We treat the case where the asymptotic expansion of $A(z_1,\ldots,z_d)$ contains only power-law type terms, and where the indices $n_1,\ldots,n_d$ tend to infinity in some polynomially stretched diagonal limit. The resulting asymptotic expansion of $a_{n_1,\ldots,n_d}$ is a sum of terms of the form \begin{equation*} I(λ_1,\ldots,λ_d) \cdot n_0^{-Θ} \cdot ρ_1^{-n_1}\cdots ρ_d^{-n_d}, \end{equation*} where $(λ_1,\ldots,λ_d) \in (0,\infty)^d$ is the direction vector of the stretched diagonal limit for $(n_1,\ldots,n_d)$, the parameter $n_0$ tends to $\infty$ at similar speed as $n_1,\ldots,n_d$, while $Θ\in \mathbb R$ and $I:(0,\infty)^d \to \mathbb C$ are determined by the asymptotic expansion of $A$.
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