[Paper Review] Asymptotic analysis of the Askey-scheme II: from Charlier to Hermite

Diego Dominici|ArXiv.org|Aug 15, 2005

Mathematical functions and polynomials22 references3 citations

TL;DR

This paper derives asymptotic approximations for Hermite polynomials $ H_n( heta) $ as $ n \to \infty $, leveraging the limit relation between Charlier and Hermite polynomials. It provides accurate, region-specific formulas involving Airy functions, Bessel functions, and trigonometric expressions, with key results valid across different ranges of $ \theta $, including oscillatory behavior in the bulk and edge scaling near $ \pm\sqrt{2n} $.

ABSTRACT

We analyze the Hermite polynomials $H_{n}(ξ)$ and their zeros asymptotically as $n o\infty,$ using the limit relation between the Charlier and Hermite polynomials. Our formulas involve some special functions and they yield very accurate approximations.

Motivation & Objective

• To derive uniform asymptotic approximations for Hermite polynomials $ H_n(\xi) $ as $ n \to \infty $, using the limiting relation between Charlier and Hermite polynomials.
• To extend asymptotic results from Charlier polynomials—previously derived in [2]—to the Hermite case via the limit $ \lim_{a\to\infty} (-1)^n (2a)^{n/2} C_n^{(a)}(a + \xi\sqrt{2a}) = H_n(\xi) $.
• To provide accurate approximations across all regions of $ \xi $, including bulk, edge, and oscillatory regimes, with explicit formulas involving special functions.
• To analyze the zeros of $ H_n(\xi) $ by transforming the zero-finding problem into a form solvable by Kepler’s equation and expressing them via Kapteyn series involving Bessel functions.

Proposed method

• Utilizes the known limit relation between Charlier and Hermite polynomials: $ \lim_{a\to\infty} (-1)^n (2a)^{n/2} C_n^{(a)}(a + \xi\sqrt{2a}) = H_n(\xi) $, to transfer asymptotic results from Charlier to Hermite polynomials.
• Applies asymptotic expansions of Charlier polynomials in six distinct regions (e.g., bulk, edges, transition zones), derived in [2], and translates them into Hermite polynomial approximations via the limit.
• Derives region-specific approximations for $ H_n(\xi) $: oscillatory in the bulk ($ |\xi| \ll \sqrt{2n} $), Airy-type near edges ($ \xi \approx \pm\sqrt{2n} $), and exponential in the tails.
• Transforms the Hermite zero-finding problem into a form resembling Kepler’s equation $ E - \varepsilon \sin E = M $, with $ \varepsilon = -2n/(2n+1) $, enabling series solutions via Bessel functions.
• Expresses the zeros $ \zeta_j^n $ as $ \sqrt{2n} \sin(\tau_j^n) $, where $ \tau_j^n $ satisfies a trigonometric equation solvable via Kapteyn series involving $ \mathrm{J}_k(k\varepsilon) $.
• Uses the reflection symmetry $ H_n(-\xi) = (-1)^n H_n(\xi) $ to relate approximations in symmetric regions and simplify expressions.

Experimental results

Research questions

• RQ1 How can asymptotic approximations for Hermite polynomials $ H_n(\xi) $ be derived uniformly across all ranges of $ \xi $ as $ n \to \infty $, especially in the bulk and edge regimes?
• RQ2 What is the precise asymptotic behavior of $ H_n(\xi) $ in the transition region near $ \xi = \pm\sqrt{2n} $, and how does it relate to the Airy function?
• RQ3 How do the zeros of $ H_n(\xi) $ distribute asymptotically, and can they be expressed in terms of special functions or series expansions?
• RQ4 Can the limit relation between Charlier and Hermite polynomials be used as a systematic method to derive asymptotics for other families in the Askey scheme?
• RQ5 What is the connection between the Hermite polynomial zeros and Kepler’s equation, and how can this be exploited to derive explicit series representations?

Key findings

• For $ \xi \ll -\sqrt{2n} $, $ H_n(\xi) \sim \Lambda_1(\xi) = \exp\left[\Phi_1(\xi)\right] U_1(\xi) $, with $ \Phi_1(\xi) = \frac{n}{2}\left[\ln(2n) - \cos(2\theta)\right] + \frac{n\pi}{2}i $, and $ \theta = \arcsin(\xi / \sqrt{2n}) $, yielding oscillatory decay.
• In the bulk region $ |\xi| \ll \sqrt{2n} $, $ H_n(\xi) \sim \Lambda_5(\xi) = \sqrt{2}(1 - \xi^2/(2n))^{-1/4} \exp\left[\frac{n}{2}(\ln(2n) - 1) + \frac{\xi^2}{2}\right] \cos(\Theta) $, where $ \Theta = \frac{\xi}{2}\sqrt{2n - \xi^2} + (n + \frac{1}{2})\arcsin(\frac{\xi}{\sqrt{2n}}) - \frac{n\pi}{2} $.
• Near the right edge $ \xi \approx \sqrt{2n} $, $ H_n(\xi) \sim \Lambda_4(\xi) = \sqrt{2\pi} n^{1/6} \exp[\Phi_4(\xi)] \mathrm{Ai}[n^{1/6}\sqrt{2}(\xi - \sqrt{2n})] $, with $ \Phi_4(\xi) = \frac{1}{2}n\ln(n/a) + x\ln(1 + \sqrt{n/a}) - \sqrt{an} - \sqrt{n} $, showing Airy-type decay.
• The zeros $ \zeta_j^n $ of $ H_n(\xi) $ satisfy $ \zeta_j^n = \sqrt{2n} \sin(\tau_j^n) $, where $ \tau_j^n $ solves a Kepler-type equation with $ \varepsilon = -2n/(2n+1) $, and is expressed as a Kapteyn series: $ \tau_j^n = \pi(1+n-2j)/(2n+1) + \sum_{k=1}^\infty \frac{1}{k} \mathrm{J}_k(-\frac{2n}{2n+1}k) \sin(2\pi(1+n-2j)k/(2n+1)) $.
• The leading-order approximation of $ \Lambda_5(\xi) $ matches formula (4.14.9) in [9], confirming consistency with established asymptotic theory: $ \Lambda_5(\xi) \sim \sqrt{2} \exp[\frac{n}{2}(\ln(2n) - 1) + \frac{\xi^2}{2}] \cos(n\pi/2 - \xi\sqrt{2n}) $.
• The method successfully transfers asymptotic results from Charlier polynomials to Hermite polynomials, yielding highly accurate approximations across all regions, including edge and bulk, with explicit dependence on special functions like Airy and Bessel functions.