The zero distribution of the Tricomi-Carlitz polynomials

In this paper, we improve our previous results concerning the asymptotic distribution of the zeros of the Tricomi-Carlitz polynomials. Our approach is a probabilistic one, motivated by the fact that the central limit theorem is simply a version of the weak law of large numbers whose distribution is a δ-function. Also, our argument requires an extension of the Grommer-Hamburger theorem. This generalization is appropriate to situations where uniform convergence on compact subsets is difficult to establish, and is accomplished by means of an approximation-theoretic tool, namely, the closure of a special set of rational functions.