A transfer theorem for multivariate Delta-analytic functions with a power-law singularity

Abstract

This paper presents a multivariate generalization of Flajolet and Odlyzko's transfer theorem. Similarly to the univariate version, the theorem assumes $\Delta$-analyticity (defined coordinate-wise) of a function $A(z_1,\dots ,z_d)$ at a unique dominant singularity $({\rho }_1,\dots ,{\rho }_d)\in ({\mathbb{C}}_{\ast }{)}^d$, and allows one to translate, on a term-by-term basis, an asymptotic expansion of $A(z_1,\dots ,z_d)$ around $({\rho }_1,\dots ,{\rho }_d)$ into a corresponding asymptotic expansion of its Taylor coefficients $a_{n_1,\dots ,n_d}$. We treat the case where the asymptotic expansion of $A(z_1,\dots ,z_d)$ contains only power-law type terms, and where the indices $n_1,\dots ,n_d$ tend to infinity in some polynomially stretched diagonal limit. The resulting asymptotic expansion of $a_{n_1,\dots ,n_d}$ 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 $({\lambda }_1,\dots ,{\lambda }_d)\in (0,\infty {)}^d$ is the direction vector of the stretched diagonal limit for $(n_1,\dots ,n_d)$, the parameter $n_0$ tends to $\infty$ at similar speed as $n_1,\dots ,n_d$, while $\Theta \in \mathbb{R}$ and $I:(0,\infty {)}^d\to \mathbb{C}$ are determined by the asymptotic expansion of $A$.

23 pages, 2 figures, preliminary version