Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients

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 $μ-γ_n$, provin