Editing Double factorial (section)

===Alternative extension of the multifactorial===
Alternatively, the multifactorial {{math|''z''!<sub>(''α'')</sub>}} can be extended to most real and complex numbers {{mvar|z}} by noting that when {{mvar|z}} is one more than a positive multiple of the positive integer {{mvar|α}} then

<math display="block">\begin{align} z!_{(\alpha)} &= z(z-\alpha)\cdots (\alpha+1) \\
&= \alpha^\frac{z-1}{\alpha}\left(\frac{z}{\alpha}\right)\left(\frac{z-\alpha}{\alpha}\right)\cdots \left(\frac{\alpha+1}{\alpha}\right) \\
&= \alpha^\frac{z-1}{\alpha} \frac{\Gamma\left(\frac{z}{\alpha}+1\right)}{\Gamma\left(\frac{1}{\alpha}+1\right)}\,.
\end{align}</math>

This last expression is defined much more broadly than the original.  In the same way that {{math|''z''!}} is not defined for negative integers, and {{math|''z''‼}} is not defined for negative even integers, {{math|''z''!<sub>(''α'')</sub>}} is not defined for negative multiples of {{mvar|α}}.  However, it is defined and satisfies {{math|1=(''z''+''α'')!<sub>(''α'')</sub> = (''z''+''α'')·''z''!<sub>(''α'')</sub>}} for all other complex numbers&nbsp;{{mvar|z}}. This definition is consistent with the earlier definition only for those integers {{mvar|z}} satisfying&nbsp;{{math|''z'' ≡ 1 mod ''α''}}.

In addition to extending {{math|''z''!<sub>(''α'')</sub>}} to most complex numbers&nbsp;{{mvar|z}}, this definition has the feature of working for all positive real values of&nbsp;{{mvar|α}}. Furthermore, when {{math|1=''α'' = 1}}, this definition is mathematically equivalent to the {{math|Π(''z'')}} function, described above. Also, when {{math|1=''α'' = 2}}, this definition is mathematically equivalent to the [[#Complex arguments|alternative extension of the double factorial]].