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 $A(z_1,\ldots,z_d)$ at a unique dominant singularity $(\rho_1,\ldots,\rho_d) \in (\mathbb C_*)^d$, and allows one to translate, on a term-by-term basis, an asymptotic expansion of $A(z_1,\ldots,z_d)$ around $(\rho_1,\ldots,\rho_d)$ 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(\lambda_1,\ldots,\lambda_d) \cdot n_0^{-\Theta} \cdot \rho_1^{-n_1}\cdots \rho_d^{-n_d}, \end{equation*} where $(\lambda_1,\ldots,\lambda_d) \in (0,\infty)^d$ 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 $\Theta\in \mathbb R$ and $I:(0,\infty)^d \to \mathbb C$ are determined by the asymptotic expansion of $A$.