Difference between revisions of "Rising factorial"
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The rising factorial is given by | The rising factorial is given by | ||
$$a^{\overline{\xi}} = \dfrac{\Gamma(a+\xi)}{\Gamma(a)}.$$ | $$a^{\overline{\xi}} = \dfrac{\Gamma(a+\xi)}{\Gamma(a)}.$$ | ||
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| + | =Properties= | ||
| + | {{:Rodrigues formula for Meixner polynomial}} | ||
Revision as of 15:32, 20 May 2015
The rising factorial is given by $$a^{\overline{\xi}} = \dfrac{\Gamma(a+\xi)}{\Gamma(a)}.$$
Contents
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.
