Relationship between Meixner polynomials and Charlier polynomials

Theorem: The following formula holds: $$\displaystyle\lim_{\beta \rightarrow \infty} M_n \left(x;\beta,\dfrac{a}{\beta+a} \right) = C_n(x;a),$$ where $M_n$ denotes a Meixner polynomial and $C_n$ denotes a Charlier polynomial.

Proof: █