The double factorial of a positive integer is a generalization of the usual factorial
defined by
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(1)
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Note that ,
by definition (Arfken 1985, p. 547).
The origin of the notation appears not to not be widely known and is not mentioned
in Cajori (1993).
For ,
1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in
for
, 1, ... are 1, 4, 80, 1285, 17831, 228289, 2782857, 32828532,
... (OEIS A114488).
The double factorial is implemented in the Wolfram Language as n!! or Factorial2[n].
The double factorial is a special case of the multifactorial.
The double factorial can be expressed in terms of the gamma function by
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(2)
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(Arfken 1985, p. 548).
The double factorial can also be extended to negative odd integers using the definition
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(3)
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(4)
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for ,
1, ... (Arfken 1985, p. 547).
Similarly, the double factorial can be extended to complex arguments as
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(5)
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There are many identities relating double factorials to factorials. Since
![(2n+1)!!2^nn!
=[(2n+1)(2n-1)...1][2n][2(n-1)][2(n-2)]...2·1
=[(2n+1)(2n-1)...1][2n(2n-2)(2n-4)...2]
=(2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4)...2·1
=(2n+1)!,](/images/equations/DoubleFactorial/NumberedEquation4_400.svg) |
(6)
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it follows that .
For
,
1, ..., the first few values are 1, 3, 15, 105, 945, 10395, ... (OEIS A001147).
Also, since
|
(7)
| |||
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(8)
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(9)
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it follows that .
For
,
1, ..., the first few values are 1, 2, 8, 48, 384, 3840, 46080, ... (OEIS A000165).
Finally, since
![(2n-1)!!2^nn!
=[(2n-1)(2n-3)...1][2n][2(n-1)][2(n-2)]...2(1)
=(2n-1)(2n-3)...1][2n(2n-2)(2n-4)...2]
=2n(2n-1)(2n-2)(2n-3)(2n-4)...2(1)
=(2n)!,](/images/equations/DoubleFactorial/NumberedEquation5_400.svg) |
(10)
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it follows that
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(11)
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For odd,
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(12)
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(13)
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(14)
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For even,
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(15)
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(16)
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(17)
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Therefore, for any ,
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(18)
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(19)
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The double factorial satisfies the beautiful series
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(20)
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(21)
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(22)
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The latter gives rhe sum of reciprocal double factorials in closed form as
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(23)
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(24)
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(25)
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(OEIS A143280), where is a lower incomplete
gamma function. This sum is a special case of the reciprocal
multifactorial constant.
A closed-form sum due to Ramanujan is given by
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(26)
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(Hardy 1999, p. 106). Whipple (1926) gives a generalization of this sum (Hardy 1999, pp. 111-112).