Central Limit Theorems for Combinatorial Numbers Associated with Laguerre Polynomials

In this paper, we study limit theorems for numbers satisfying a class of triangular arrays, which are defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain analytical expressions for the semi-exponential generating function of several classes of the numbers, including combinatorial numbers associated with Laguerre polynomials. We apply these results to prove the numbers' asymptotic normality and specify the convergence rate to the limiting distribution.