The double factorial of a positive integer is a generalization of the usual factorial defined by
(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
(2)
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(Arfken 1985, p. 548).
The double factorial can also be extended to negative odd integers using the definition
(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
(5)
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There are many identities relating double factorials to factorials. Since
(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
(10)
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it follows that
(11)
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For odd,
(12)
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(13)
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(14)
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For even,
(15)
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(16)
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(17)
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Therefore, for any ,
(18)
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(19)
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The double factorial satisfies the beautiful series
(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
(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
(26)
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(Hardy 1999, p. 106). Whipple (1926) gives a generalization of this sum (Hardy 1999, pp. 111-112).