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Rational approximation of Euler's constant using multiple orthogonal polynomials

Thomas Wolfs, Walter Van Assche

TL;DR

This work advances rational approximation methods for Euler's constant $\\gamma$ and the Gompertz constant $\\delta$ by employing mixed-type multiple orthogonal polynomials tied to the exponential integral, recasting Rivoal's construction via type I Laguerre polynomials and a Riemann-Hilbert framework to streamline proofs and analysis. The authors obtain new approximants with improved Diophantine and asymptotic properties, achieving sub-exponential error decay with rates $\\sim 4/(n^{1/4}\\ln n)$ and explicit integer-denominator structures, and they extend the dual framework to the Gompertz constant. A rigorous connection between the type I and II constructions is established through Mahler/RH relations, clarifying the landscape of Hermite-Pad\\é approximation in this context. Finally, the paper outlines a path to further improvements by applying higher-order differential operators, potentially yielding even faster decay of the approximation error. These results deepen the link between mixed-type orthogonal polynomials, irrationality questions, and explicit Diophantine constructions in special constants.

Abstract

We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the exponential integral. The dual family of multiple orthogonal polynomials leads to new rational approximants of the Gompertz constant that improve those of Aptekarev et al. (2007). Our approach is motivated by the fact that we can reformulate Rivoal's construction in terms of type I multiple Laguerre polynomials of the first kind by making use of the underlying Riemann-Hilbert problem. As a consequence, we can drastically simplify Rivoal's approach, which allows us to study the Diophantine and asymptotic properties of the approximants more easily.

Rational approximation of Euler's constant using multiple orthogonal polynomials

TL;DR

This work advances rational approximation methods for Euler's constant and the Gompertz constant by employing mixed-type multiple orthogonal polynomials tied to the exponential integral, recasting Rivoal's construction via type I Laguerre polynomials and a Riemann-Hilbert framework to streamline proofs and analysis. The authors obtain new approximants with improved Diophantine and asymptotic properties, achieving sub-exponential error decay with rates and explicit integer-denominator structures, and they extend the dual framework to the Gompertz constant. A rigorous connection between the type I and II constructions is established through Mahler/RH relations, clarifying the landscape of Hermite-Pad\\é approximation in this context. Finally, the paper outlines a path to further improvements by applying higher-order differential operators, potentially yielding even faster decay of the approximation error. These results deepen the link between mixed-type orthogonal polynomials, irrationality questions, and explicit Diophantine constructions in special constants.

Abstract

We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the exponential integral. The dual family of multiple orthogonal polynomials leads to new rational approximants of the Gompertz constant that improve those of Aptekarev et al. (2007). Our approach is motivated by the fact that we can reformulate Rivoal's construction in terms of type I multiple Laguerre polynomials of the first kind by making use of the underlying Riemann-Hilbert problem. As a consequence, we can drastically simplify Rivoal's approach, which allows us to study the Diophantine and asymptotic properties of the approximants more easily.
Paper Structure (9 sections, 20 theorems, 132 equations)

This paper contains 9 sections, 20 theorems, 132 equations.

Table of Contents

  1. Introduction and overview of the results
  2. Approximants for Euler's constant
  3. Motivation
  4. Quality
  5. Approximants for the Gompertz constant
  6. Motivation
  7. Quality
  8. Connection
  9. Further improvement

Key Result

Lemma 1.1

Let $x\in\mathbb{R}$ and suppose that there exists a sequence $(p_n,q_n)_{n\in\mathbb{N}}$ in $\mathbb{Z}\times\mathbb{N}$ such that $r(n) = - \ln \left| x - p_n/q_n \right| / \ln q_n \in\mathbb{R}$ for all $n$. If $\liminf_{n\to\infty} r(n) > 1$, then $x$ is irrational.

Theorems & Definitions (32)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 22 more