Padé approximations for products of functions
Makoto Kawashima
TL;DR
The paper solves the problem of constructing explicit Padé approximations for the product of binomial functions and logarithmic powers, via a formal, exponential- transform approach that leverages Mahler’s framework. This yields new quantitative linear independence results for linear forms in $$(1+\alpha)^{\omega_i}\log^{j_i}(1+\alpha)$$ across complex and $p$-adic settings, with fully explicit constants and conditions. A key methodological advance is the transfer of exponential Padé approximants to the product setting, plus a determinant-based non-vanishing argument and a linear independence criterion that culminate in the main Diophantine estimate. An auxiliary result shows that Padé approximation of a single polylogarithm is generally perfect, which is consolidated in the appendix with a Laurent-series treatment. The work thus advances both the construction of Padé approximants for composite functions and the arithmetic applications in transcendence/irrationality problems, including polylogarithm cases.
Abstract
In this article, we construct new Padé approximations for the \emph{product} of binomial functions and powers of logarithmic functions. While several explicit Padé approximants are known for powers of exponential functions, binomial functions, and logarithmic functions individually, an explicit Padé construction for the product of these functions has not yet been directly achieved. Our main result yields arithmetic applications, providing new linear independence measures for linear forms in $(1+α)^{ω_i}\log^{j_i}(1+α)$ for $1 \le i \le m$ and $0 \le j_i \le r_i - 1$, where $0 < m, r_1, \ldots, r_m \in \mathbb{Z}_{\geq 1}$, $ω_1, \ldots, ω_m \in \mathbb{Q}$, and $0 \le ω_1 < \cdots < ω_m < 1$. These results hold with algebraic coefficients in both the complex and $p$-adic cases. Additionally, we establish that Padé approximation of a single polylogarithm is, in general, perfect.