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The double factorial of a positive integer is a generalization of the usual factorial defined by
(1) |
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) |
(Arfken 1985, p. 548).
The double factorial can also be extended to negative odd integers using the definition
(3) |
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(4) |
for , 1, ... (Arfken 1985, p. 547).
Similarly, the double factorial can be extended to complex arguments as
(5) |
There are many identities relating double factorials to factorials. Since
(6) |
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) |
it follows that . For , 1, ..., the first few values are 1, 2, 8, 48, 384, 3840, 46080, ... (OEIS A000165).
Finally, since
(10) |
it follows that
(11) |
For odd,
(12) |
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(13) |
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(14) |
For even,
(15) |
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(16) |
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(17) |
Therefore, for any ,
(18) |
(19) |
The double factorial satisfies the beautiful series
(20) |
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(21) |
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(22) |
The latter gives rhe sum of reciprocal double factorials in closed form as
(23) |
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(24) |
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(25) |
(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) |
(Hardy 1999, p. 106). Whipple (1926) gives a generalization of this sum (Hardy 1999, pp. 111-112).
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 544-545 and 547-548, 1985.
Cajori, F. A History of Mathematical Notations, Vol. 2. New York: Dover, 1993.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
Meserve, B. E. "Double Factorials." Amer. Math. Monthly 55, 425-426, 1948.
Sloane, N. J. A. Sequences A000165/M1878, A001147/M3002, A006882/M0876, A114488, and A143280 in "The On-Line Encyclopedia of Integer Sequences."
Whipple, F. J. W. "On Well-Poised Series, Generalised Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247-263, 1926.
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