Variations of Central Limit Theorems and Stirling numbers of the First Kind

We construct a new parametrization of double sequences $\{A_{n,k}(s){\}}_{n,k}$ between $A_{n,k}(0)=\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ and $A_{n,k}(1)=\frac{1}{n!}\text{stirl}nk$, where $\text{stirl}nk$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de,Moivre–Laplace central limit theorem and Goncharov’s result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.