Pad\'{e} approximations for products of functions

TL;DR

This paper develops new Padé approximations for products of binomial and logarithmic functions, leading to improved measures of linear independence and insights into polylogarithm approximations in complex and p-adic contexts.

Contribution

The paper introduces explicit Padé approximations for products of binomial and logarithmic functions, a novel achievement with applications in number theory and approximation theory.

Findings

Provides new linear independence measures for specific linear forms.

Establishes that Padé approximation of a single polylogarithm is generally perfect.

Extends approximation techniques to complex and p-adic cases.

Abstract

In this article, we construct new Pad\'{e} approximations for the \emph{product} of binomial functions and powers of logarithmic functions. While several explicit Pad\'{e} approximants are known for powers of exponential functions, binomial functions, and logarithmic functions individually, an explicit Pad\'{e} 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 $(1+\alpha)^{\omega_i}\log^{j_i}(1+\alpha)$ for $1 \le i \le m$ and $0 \le j_i \le r_i - 1$, where $0 < m, r_1, \ldots, r_m \in \mathbb{Z}_{\geq 1}$, $\omega_1, \ldots, \omega_m \in \mathbb{Q}$, and $0 \le \omega_1 < \cdots < \omega_m < 1$. These results hold with algebraic coefficients in both the complex and $p$-adic cases. Additionally, we establish that Pad\'{e} approximation of a single polylogarithm is, in general, perfect.