Meixner polynomials

Remarks

They are orthogonal on the nonnegative integers: $$\sum_{x=0}^{\infty} \frac{(\beta)_x}{x!} c^x M_m(x; \beta, c) M_n(x; \beta, c) = \frac{c^{-n} \, n!}{(\beta)_n (1 - c)^\beta} \delta_{mn}.$$ Example: $M_1(x; \beta, c) = 1 - \frac{(1 - c) x}{\beta c}$. Check: the weights sum to $(1 - c)^{-\beta}$, and their mean value of $x$ is $\frac{\beta c}{1 - c}$, so the weighted sum of $M_1$ is $(1 - c)^{-\beta} (1 - 1) = 0$. Non-example: for $c \ge 1$ the clause $0 < c < 1$ fails, and the weights $\frac{(\beta)_x}{x!} c^x$ do not have a finite sum, so the orthogonality relation has no meaning. Limit to Laguerre: $\lim_{c \to 1} M_n\left( \frac{x}{1 - c}; \alpha + 1, c \right) = \frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}$.