AI 中文总结
该论文研究广义Charlier多项式递推系数的完全阶乘渐近，通过参数化归结为离散Painlevé II方程，结合Bessel比较与Hankel行列式扇区分解，精确得到连接公式和渐近，并揭示可积结构。
AI 中文摘要
我们研究了与离散权重\\[ \rho_\mu(k) = \frac{\mu^k}{(k!)^2}, \qquad k\in\mathbb N_0, \qquad \mu>0. \\]相关的广义Charlier多项式的首一递推系数。我们建立了偏差$\beta_n-n$和$\mu-\gamma_n$的完全阶乘渐近，证明它们超越所有代数阶，并出现在尺度$\mu^{n}/(n!)^2$上。关键步骤是参数化$\beta_n-n=\sqrt{\mu}\\,c_nc_{n+1}$，$\gamma_n=\mu(1-c_n^2)$，这将问题归结为一个满足离散Painlevé II方程的正序列$(c_n)$。在$c_n=0$附近线性化得到离散Bessel方程，其劣解为$J_n(2\sqrt{\mu})$。我们证明比值$c_n/J_n(2\sqrt{\mu})$收敛到有限极限$L(\mu)$，并通过结合Bessel比较与基于相关Hankel行列式的扇区分解的全局分析，精确识别出$L(\mu)=1$。这给出了精确的连接公式\\[ c_n \sim J_n(2\sqrt{\mu}), \qquad n\to\infty, \\]从而得到精确渐近\\[ \beta_n-n\sim \frac{\mu^{n+1}}{n!(n+1)!}, \qquad \mu-\gamma_n\sim \frac{\mu^{n+1}}{(n!)^2}. \\]我们进一步发展了一种基于Hankel行列式解释为离散对数气体的补充方法。我们证明Hankel行列式分解为$\Delta_n(\mu)=\mu^{n(n-1)/2}T_n(\mu)$，并允许扇区分解$T_n(\mu)=1+\sum_{q=1}^{n}S_q(\mu)$，其中扇区贡献与$n$无关。我们建立了$S_q(\mu)\sim e^\mu\mu^q/(q!)^2$，提供了阶乘修正的独立推导。综合来看，Bessel比较和扇区分解揭示了丰富的可积结构，将离散Painlevé动力学、Toda变形和Hankel-配分表示联系起来。
英文摘要
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 $β_n-n$ 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 $β_n-n=\sqrtμ\,c_nc_{n+1}$, $γ_n=μ(1-c_n^2)$, 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 $J_n(2\sqrtμ)$. We prove that the ratio $c_n/J_n(2\sqrtμ)$ 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 \[ c_n \sim J_n(2\sqrtμ), \qquad n\to\infty, \] 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 $Δ_n(μ)=μ^{n(n-1)/2}T_n(μ)$ and admits a sector decomposition $T_n(μ)=1+\sum_{q=1}^{n}S_q(μ)$ with sector contributions independent of $n$. We establish $S_q(μ)\sim e^μμ^q/(q!)^2$, 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.
