Rising factorial

The rising factorial is given by $$a^{\overline{\xi}} = \dfrac{\Gamma(a+\xi)}{\Gamma(a)}.$$

Properties

Theorem

The following formula holds: $$\dfrac{\beta^{\overline{x}}c^x}{x!} M_n(x;\beta,c)= \nabla^n \left[ \dfrac{(\beta+n)^{\overline{x}}}{x!}c^x \right],$$ where $\nabla$ denotes the backwards difference operator $\nabla f = f(x)-f(x-1)$, $\beta^{\overline{x}}$ denotes a rising factorial and $M_n$ is a Meixner polynomial.