On the Transition of Charlier Polynomials to the Hermite Function

Abstract It has been known for over 70 years that there is an asymptotic transition of Charlier polynomials to Hermite polynomials. This transition, which still presented in its classical form modern reference works, valid if and only a certain parameter integer. In this light, it surprising much more powerful exists from the function , any real value parameter. greatly strengthens connections between special functions, with applications queueing theory, where crucial solving first-passage problems moving boundaries. shown paper convergence locally uniform, sharp rate bound proved. addition, derivatives derivative function, again bound. Finally, proved zeros converge function.