On local limit theorems for

In this work, we establish local limit theorems for q- multinomial distributions of the first and second kind and of their discrete limits multiple Heine and multiple Euler distributions respectively. Specifically, the pointwise convergence of the q-multinomial distribution of the first kind, as well as for its discrete limit, the multiple Heine distribution, to a multivariate Stieltjes–Wigert type distribution, are provided. Moreover, the pointwise convergence of the q-multinomial distribution of the second kind, as well as for its discrete limit, the multiple Euler distribution, to a multivariate deformed Gaussian distribution, are proved. Interesting applications of the asymptotic behaviour of q-multinomials distributions of the two kinds are presented.