[Paper Review] Convergence in $\C$ of series for the Lambert $W$ Function

G. A. Kalugin, David J. Jeffrey|arXiv (Cornell University)|Aug 2, 2012

Sports Dynamics and Biomechanics9 references3 citations

TL;DR

This paper establishes the precise domains of convergence for asymptotic series expansions of the Lambert W function in both real and complex domains. It introduces a parameterized invariant transformation that extends convergence domains at the cost of reduced convergence speed, derives asymptotic expressions for expansion coefficients, and reveals new combinatorial identities through connections between Stirling numbers, Eulerian numbers, and the Omega constant.

ABSTRACT

We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including asymptotic expressions for the expansion coefficients. We introduce an new invariant transformation of the series. The transformation contains a parameter whose effect on the domain and rate of convergence is studied theoretically and numerically. We also give alternate representations of the expansion coefficients, which imply a number of combinatorial identities.

Motivation & Objective

• To determine the exact domains of convergence for known asymptotic series expansions of the Lambert W function in the complex plane.
• To analyze the convergence behavior of series (1.3) and (1.5) in both real and complex settings, identifying why (1.5) has a wider domain.
• To introduce and study a parameterized invariant transformation that modifies convergence properties without altering the series structure.
• To derive asymptotic expressions for expansion coefficients in the series (1.5) and (1.6), and to relate them to combinatorial numbers such as Stirling and Eulerian numbers.
• To uncover new combinatorial identities, including generalizations of the Carlitz-Riordan identities, through equivalent representations of the expansion coefficients.

Proposed method

• Derives the domain of convergence for series (1.3) using the implicit function theorem and singular point analysis in the complex σ-plane, identifying critical points via the Lambert W function.
• Applies the implicit function theorem to the fundamental relation (1.4) to express the radius of convergence as the distance to the nearest singular point in the complex σ-plane.
• Introduces a parameterized invariant transformation of the series that preserves structure while altering convergence behavior, with theoretical and numerical analysis of the parameter’s effect.
• Establishes asymptotic estimates for expansion coefficients in series (1.5) using saddle-point methods and comparison with known combinatorial identities.
• Re-expresses the coefficients in series (1.6) in three new forms—via associated Stirling numbers of the first kind, 2-associated Stirling numbers of the second kind, and second-order Eulerian numbers—enabling new combinatorial identities.
• Uses the Omega constant ω₀ = W(1) as a key parameter to connect different series representations and derive identities involving alternating sums of Stirling and Eulerian numbers.

Experimental results

Research questions

• RQ1 What are the exact domains of convergence for the asymptotic series expansions of the Lambert W function in the complex plane?
• RQ2 How does the parameter in the invariant transformation affect the convergence domain and rate of convergence of the series?
• RQ3 What are the asymptotic expressions for the expansion coefficients in the series (1.5) and (1.6), and how do they relate to known combinatorial numbers?
• RQ4 Can equivalent representations of the expansion coefficients lead to new combinatorial identities, such as generalizations of the Carlitz-Riordan identities?
• RQ5 Why does the series (1.5) exhibit a significantly wider domain of convergence than (1.3), and what is the underlying mathematical reason?

Key findings

• The domain of convergence for series (1.3) is precisely defined by the inequality involving the real part of the Lambert W function: ln|σ| < 1 − τ/σ + Re W₋₁(−e^{τ/σ−1}).
• The series (1.5) has a significantly wider domain of convergence than (1.3) in both real and complex domains, due to the structure of its coefficient expansion and the use of 2-associated Stirling numbers.
• The parameterized invariant transformation extends the domain of convergence with increasing parameter p, but reduces the rate of convergence, making the minimal p value optimal within overlapping convergence regions.
• Asymptotic estimates for the coefficients in (1.5) are derived as ∑_{p=1}^{m−1} S₂(p+m−1, p) ∼ (m−1)! / [2√(πm)(2ln2−1)^{m−1/2}] as m→∞, consistent with known results.
• The expansion coefficients in (1.6) are shown to be equivalent to sums involving second-order Eulerian numbers, leading to new identities such as ∑_{k=0}^{n−1} ⟨⟨n−1|k⟩⟩ (−1)^k ω₀^k / (1+ω₀)^{n−1} = ∑_{k=0}^{n−1} (−1)^{n+k−1} d(n+k−1,k) ω₀^k / (1+ω₀)^k.
• The paper confirms the Carlitz-Riordan identity and derives a new identity for alternating sums of 2-associated Stirling numbers of the second kind: ∑_{p=0}^{m−1} (−1)^{p+m−1} S₂(p+m−1,p) = (m−1)! at σ=0.