Asymptotic normality of the Stirling-Whitney-Riordan triangle

Abstract

Recently, Zhu \cite{B.-X.Z21} introduced a Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k\geq0}$ satisfying the recurrence:
\begin{eqnarray*}T_{n,k}=&&(b_{1}k+b_{2})T_{n-1,k-1}+[(2\lambda
b_{1}+a_{1})k+a_{2}+\lambda(b_{1}+b_{2})]T_{n-1,k}+
\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda(a_{1}+\lambda b_{1})(k+1)T_{n-1,k+1},
\end{eqnarray*}
where initial conditions $T_{n,k}=0$ unless $0\leq k\leq n$ and
$T_{0,0}=1$.
Denote by $T_n=\sum_{k=0}^nT_{n,k}$.
In this paper, we show the asymptotic normality of
$T_{n,k}$ and give an asymptotic formula of $T_n$.
As applications,
we show the asymptotic normality of many famous combinatorial numbers,
such as the Stirling numbers of the second kind,
the Whitney numbers of the second kind, the $r$-Stirling numbers and the $r$-Whitney numbers of the second kind.