Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients
Abstract
We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight ρμ(k) = μk/(k!)2, k∈ℕ0, μ>0. We establish complete factorial asymptotics for the deviations βn−n and μ−γn, proving that they are beyond all algebraic orders and appear at the scale μn/(n!)2. The key step is a parametrization βn−n=√μ cncn+1, γn=μ(1−cn2), which reduces the problem to a positive sequence (cn) satisfying a discrete Painlevé II equation. Linearization around cn=0 yields a discrete Bessel equation whose recessive solution is Jn(2√μ). We prove that the ratio cn/Jn(2√μ) converges to a finite limit L(μ), which we identify exactly as L(μ)=1 by combining Bessel comparison with a global analysis based on the sector decomposition of the associated Hankel determinant. This yields the exact connection formula cn ∼ Jn(2√μ), n→∞, and consequently the precise asymptotics βn−n∼ (μn+1)/(n!(n+1)!), μ−γn∼ (μn+1)/(n!)2. We further develop a complementary approach based on Hankel determinants interpreted as a discrete logarithmic gas. We show that the Hankel determinant factorizes as Δn(μ)=μn(n−1)/2Tn(μ) and admits a sector decomposition Tn(μ)=1+∑q=1nSq(μ) with sector contributions independent of n. We establish Sq(μ)∼ eμμq/(q!)2, providing an independent derivation of the factorial corrections. Together, the Bessel comparison and sector decomposition reveal a rich integrable structure linking discrete Painlevé dynamics, Toda deformations, and Hankel--partition representations.
Related
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BibTeX
@misc{hounkonnou2026complete,
title = {{Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients}},
author = {Mahouton Norbert Hounkonnou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00396},
eprint = {2610.00396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00396},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients
AU - Mahouton Norbert Hounkonnou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00396
UR - https://arxiv.org/abs/2610.00396
AB - We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight ρ_μ(k) = μ^k/(k!)^2, k∈ℕ_0, μ>0. We establish complete factorial asymptotics for the deviations β_n−n and μ−γ_n, proving that they are beyond all algebraic orders and appear at the scale μ^n/(n!)^2. The key step is a parametrization β_n−n=√μ c_nc_n+1, γ_n=μ(1−c_n^2), which reduces the problem to a positive sequence (c_n) satisfying a discrete Painlevé II equation. Linearization around c_n=0 yields a discrete Bessel equation whose recessive solution is J_n(2√μ). We prove that the ratio c_n/J_n(2√μ) converges to a finite limit L(μ), which we identify exactly as L(μ)=1 by combining Bessel comparison with a global analysis based on the sector decomposition of the associated Hankel determinant. This yields the exact connection formula c_n ∼ J_n(2√μ), n→∞, and consequently the precise asymptotics β_n−n∼ (μ^n+1)/(n!(n+1)!), μ−γ_n∼ (μ^n+1)/(n!)^2. We further develop a complementary approach based on Hankel determinants interpreted as a discrete logarithmic gas. We show that the Hankel determinant factorizes as Δ_n(μ)=μ^n(n−1)/2T_n(μ) and admits a sector decomposition T_n(μ)=1+∑_q=1^nS_q(μ) with sector contributions independent of n. We establish S_q(μ)∼ e^μμ^q/(q!)^2, providing an independent derivation of the factorial corrections. Together, the Bessel comparison and sector decomposition reveal a rich integrable structure linking discrete Painlevé dynamics, Toda deformations, and Hankel--partition representations.
ER -