  It has been known for over 70 years that there is an asymptotic transition of
Charlier polynomials to Hermite polynomials. This transition, which is still
presented in its classical form in modern reference works, is valid if and only
if a certain parameter is integer. In this light, it is surprising that a much
more powerful transition exists from Charlier polynomials to the Hermite
function, valid for any real value of the parameter. This greatly strengthens
the asymptotic connections between Charlier polynomials and special functions,
with applications for instance in queueing theory.
  It is shown in this paper that the convergence is uniform in bounded
intervals, and a sharp rate bound is proved. It is also shown that there is a
transition of derivatives of Charlier polynomials to the derivative of the
Hermite function, again with a sharp rate bound. Finally, it is proved that
zeros of Charlier polynomials converge to zeros of the Hermite function. While
rigorous, the proofs use only elementary techniques.
