[Paper Review] Mathieu's series: inequalities, asymptotics and positive definiteness

В. П. Заставный|ArXiv.org|Jan 8, 2009

Functional Equations Stability Results3 citations

TL;DR

This paper establishes sharp inequalities, asymptotic expansions, and connections to positive definite and completely monotonic functions for generalized Mathieu series. It derives exact bounds using integral representations, Laplace transforms, and properties of Bernoulli and Euler polynomials, proving that the generalized Mathieu series is positive definite under specific parameter constraints, with exact thresholds derived via the Hausdorff-Bernstein-Widder theorem and Hankel transforms.

ABSTRACT

Inequalities, asymptotics and, for some specific cases, asymptotical expansions were obtained for generalized Mathieu's series. A connection between inequalities for Mathieu's series and positive definite and completely monotonic functions.

Motivation & Objective

• To derive sharp inequalities for generalized Mathieu series with parameters γ, α, μ, u, t.
• To establish asymptotic expansions for the series as t → ∞ using Bernoulli and Euler polynomials.
• To investigate the connection between Mathieu-type series and positive definite functions via Hankel and Fourier transforms.
• To determine exact conditions under which the generalized series is completely monotonic or positive definite.
• To resolve open problems on optimal constants in double inequalities for the classical Mathieu series.

Proposed method

• Derivation of integral representations using the Laplace transform of x/(e^x - 1), leading to expressions involving F(x) = x/(e^x - 1).
• Application of Polya’s theorem on completely monotonic functions to the derivative of F(x), ensuring non-negativity of cosine transforms.
• Use of the Hausdorff-Bernstein-Widder theorem to characterize positive definite functions via Laplace transforms of non-negative measures.
• Employment of Hankel transforms and their connection to Fourier transforms of radial functions to analyze positive definiteness.
• Derivation of asymptotic expansions via the Euler-Maclaurin formula and Bernoulli number series in inverse powers of t.
• Establishment of inequalities through comparison with difference series and telescoping bounds involving Riemann zeta functions.

Experimental results

Research questions

• RQ1 What are the optimal constants a and b such that 1/(t² + a) < S(t,u,γ,α,μ) < 1/(t² + b) for generalized Mathieu series?
• RQ2 Under what conditions is the generalized Mathieu series positive definite on (0, ∞)?
• RQ3 How do the asymptotic expansions of the series behave as t → ∞, and what is the structure of the coefficients?
• RQ4 What is the exact relationship between the positivity of the series and the complete monotonicity of associated functions?
• RQ5 Can the sharp bounds for the classical Mathieu series be derived from generalized parameter families?

Key findings

• The optimal lower bound constant is a = 1/(2ζ(3)) and the optimal upper bound is b = 1/6 for the classical Mathieu series, as proven by Alzer, Brenner, and Ruehr.
• The generalized Mathieu series S(t,u,γ,α,μ) is completely monotonic if and only if δ = α(μ+1) - γ > 1, ensuring convergence and positivity of derivatives.
• The function ψ_{p,u,μ}(t) is completely monotonic on (0, ∞) if and only if 0 ≤ p ≤ m_∞(u), where m_∞(u) = inf_{x>0} (-ln φ_u(x)/x) and φ_u(x) is a generating function related to the series.
• The exact threshold for positive definiteness of g_{p,u}(r) = e^{-p r} - φ_u(r) is p ≤ √m_∞(u), derived via Schoenberg’s theorem and Hankel transform analysis.
• Asymptotic expansion of the series as t → ∞ is given by ∑_{k=0}^∞ (-1)^k B_{2k} / t^{2k+2}, where B_{2k} are Bernoulli numbers.
• The inequality ∑_{k=1}^∞ k/(k² + t²)^3 < (S(t))² holds for t ≥ 0, confirming a conjecture by Alzer and others.