Editing Double factorial (section)

==Extensions==

===Negative arguments===
The ordinary factorial, when extended to the [[gamma function]], has a [[Pole (complex analysis)|pole]] at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its [[recurrence relation]]
<math display="block">n!! = n \times (n-2)!!</math>
to give
<math display="block">n!! = \frac{(n+2)!!}{n+2}\,.</math>
Using this inverted recurrence, (−1)!!&nbsp;=&nbsp;1, (−3)!!&nbsp;=&nbsp;−1, and (−5)!!&nbsp;=&nbsp;{{sfrac|1|3}}; negative odd numbers with greater magnitude have fractional double factorials.<ref name="callan"/>  In particular, when {{mvar|n}} is an odd number, this gives
<math display="block">(-n)!! \times n!! = (-1)^\frac{n-1}{2} \times n\,.</math>

===Complex arguments===
Disregarding the above definition of {{math|''n''!!}} for even values of&nbsp;{{mvar|n}}, the double factorial for odd integers can be extended to most real and complex numbers {{mvar|z}} by noting that when {{mvar|z}} is a positive odd integer then<ref>{{cite book|title=Mathematical Methods: For Students of Physics and Related Fields|series=[[Undergraduate Texts in Mathematics]]|first=Sadri|last=Hassani|publisher=Springer|year=2000|isbn=9780387989587|page=266|url=https://books.google.com/books?id=dxSOzeLMij4C&pg=PA266}}</ref><ref>{{cite web|title=Double factorial: Specific values (formula 06.02.03.0005) |publisher=Wolfram Research|date=2001-10-29 |url=http://functions.wolfram.com/06.02.03.0005 |access-date=2013-03-23}}</ref>

<math display="block">\begin{align}
z!!
&= z(z-2)\cdots 5 \cdot 3 \\[3mu]
&= 2^\frac{z-1}{2}\left(\frac z2\right)\left(\frac{z-2}2\right)\cdots \left(\frac{5}{2}\right) \left(\frac{3}{2}\right) \\[5mu]
&= 2^\frac{z-1}{2} \frac{\Gamma\left(\tfrac z2+1\right)}{\Gamma\left(\tfrac12+1\right)} \\[5mu]
&= \sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right) \,,\end{align}</math>
where <math>\Gamma(z)</math> is the [[gamma function]].

The final expression is defined for all complex numbers except the negative even integers and satisfies {{math|1=(''z'' + 2)!! = (''z'' + 2) · ''z''!!}} everywhere it is defined.  As with the gamma function that extends the ordinary factorial function, this double factorial function is [[logarithmically convex]] in the sense of the [[Bohr–Mollerup theorem]].  Asymptotically, <math display=inline>n!! \sim \sqrt{2 n^{n+1} e^{-n}}\,.</math>

The generalized formula <math>\sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right)</math> does not agree with the previous product formula for {{math|''z''!!}} for non-negative ''even'' integer values of&nbsp;{{mvar|z}}.  Instead, this generalized formula implies the following alternative:
<math display="block">(2k)!! = \sqrt{\frac{2}{\pi}} 2^k \Gamma\left(k+1\right) = \sqrt{ \frac{2}{\pi} } \prod_{i=1}^k (2i) \,,</math>
with the value for 0!! in this case being
{{startplainlist|indent=1}}
* <math>0!! = \sqrt{ \frac{2}{\pi} } \approx 0.797\,884\,5608\dots</math> {{OEIS|id=A076668}}.
{{endplainlist}}

Using this generalized formula as the definition, the [[Volume of an n-ball|volume]] of an {{mvar|n}}-[[dimension]]al [[hypersphere]] of radius {{mvar|R}} can be expressed as<ref>{{cite journal|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8|s2cid=120103389}}</ref>

<math display="block">V_n=\frac{2 \left(2\pi\right)^\frac{n-1}{2}}{n!!} R^n\,.</math>