Charlier polynomial
From specialfunctionswiki
The Charlier polynomials are $$C_n(x;\mu)={}_2F_0 \left(-n,-x, -\dfrac{1}{\mu} \right) = (-1)^n n! L_n^{(-1-x)} \left( - \dfrac{1}{\mu} \right),$$ where ${}_2F_1$ denotes the hypergeometric pFq and $L_n^{(\alpha)}$ denotes the associated Laguerre polynomials.
Contents
Properties
Theorem: The Charlier polynomials are orthogonal polynomials with respect to the inner product $$\langle C_n(\cdot;\mu), C_m(\cdot;\mu) \rangle = \displaystyle\sum_{k=0}^{\infty} \dfrac{\mu^k}{k!} C_n(k;\mu)C_m(k;\mu).$$
Proof: █
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.
