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 \[ ρ_μ(k) = \frac{μ^k}{(k!)^2}, \qquad k\in\mathbb N_0, \qquad μ>0. \] We establish complete factorial asymptotics for the deviations ${\beta }_n-n$ and $\mu -{\gamma }_n$, proving that they are beyond all algebraic orders and appear at the scale ${\mu }^n/(n!{)}^2$. The key step is a parametrization ${\beta }_n-n=\sqrt{\mu }{c}_n{c}_{n+1}$, ${\gamma }_n=\mu (1-c_n^2)$, 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 $J_n(2\sqrt{\mu })$. We prove that the ratio $c_n/J_n(2\sqrt{\mu })$ converges to a finite limit $L(\mu )$, which we identify exactly as $L(\mu )=1$ by combining Bessel comparison with a global analysis based on the sector decomposition of the associated Hankel determinant. This yields the exact connection formula \[ c_n \sim J_n(2\sqrtμ), \qquad n\to\infty, \] and consequently the precise asymptotics \[ β_n-n\sim \frac{μ^{n+1}}{n!(n+1)!}, \qquad μ-γ_n\sim \frac{μ^{n+1}}{(n!)^2}. \] We further develop a complementary approach based on Hankel determinants interpreted as a discrete logarithmic gas. We show that the Hankel determinant factorizes as ${\Delta }_n(\mu )={\mu }^{n(n-1)/2}{T}_n(\mu )$ and admits a sector decomposition $T_n(\mu )=1+{\sum }_{q=1}^n{S}_q(\mu )$ with sector contributions independent of $n$. We establish $S_q(\mu )\sim e^{\mu }{\mu }^q/(q!{)}^2$, 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.