论文标题
具有幂律奇异性的多元三角分析函数的转移定理
A transfer theorem for multivariate Delta-analytic functions with a power-law singularity
论文作者
论文摘要
本文提出了Flajolet和Odlyzko的转移定理的多元概括。与单变量版本类似，该定理假定$ a（z_1，\ ldots，z_d）$的$δ$分析（定义的坐标），在独特的主导性奇点$（ρ_1，\ ldots，ρ_d，ρ_d）\ in（prrantate in（primate prrance）上，并允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个人，允许一个prranise trrans trrans trrist trrans trrans trrans， $ a（z_1，\ ldots，z_d）$（ρ_1，\ ldots，ρ_d）$的渐近扩展为其泰勒系数的相应渐近扩展$ a__ {n_1，\ ldots，n_d} $。我们处理$ a（z_1，\ ldots，z_d）$的渐近扩展的情况，仅包含幂律类型术语，而索引$ n_1，\ ldots，n_d $在某些多面有拉伸的对角度限制中趋向于无限。 $ a_ {n_1，\ ldots，n_d} $的结果渐近扩展是表格\ begin {equation*} i（λ_1，\ ldots，λ_d）\ cdot n_0^{ - θ}}} { - θ}} \ cd_11^_d_1^n_1}^{ - cd_1^{ - \ end {equation*}其中$（λ_1，\ ldots，λ_d）\ in（0，\ infty）^d $是$（n_1，\ ldots，n_d）$的对角线限制的方向向量$θ\ in \ Mathbb r $和$ i：（0，\ infty）^d \ to \ mathbb c $由$ a $的渐近扩展确定。
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 $A$.
