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Double Factorial

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The double factorial of a positive integer n is a generalization of the usual factorial n! defined by

n!!={n·(n-2)...5·3·1 n>0 odd; n·(n-2)...6·4·2 n>0 even; 1 n=-1,0.
(1)

Note that -1!!=0!!=1, by definition (Arfken 1985, p. 547).

The origin of the notation n!! appears not to not be widely known and is not mentioned in Cajori (1993).

For n=0, 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in (10^n)!! for n=0, 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

Gamma(n+1/2)=((2n-1)!!)/(2^n)sqrt(pi)
(2)

(Arfken 1985, p. 548).

DoubleFactorial

The double factorial can also be extended to negative odd integers using the definition

(-2n-1)!!=((-1)^n)/((2n-1)!!)
(3)
=((-1)^n2^nn!)/((2n)!)
(4)

for n=0, 1, ... (Arfken 1985, p. 547).

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Similarly, the double factorial can be extended to complex arguments as

z!!=2^([1+2z-cos(piz)]/4)pi^([cos(piz)-1]/4)Gamma(1+1/2z).
(5)

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)!,
(6)

it follows that (2n+1)!!=((2n+1)!)/(2^nn!). For n=0, 1, ..., the first few values are 1, 3, 15, 105, 945, 10395, ... (OEIS A001147).

Also, since

(2n)!!=(2n)(2n-2)(2n-4)...2
(7)
=[2(n)][2(n-1)][2(n-2)]...2
(8)
=2^nn!,
(9)

it follows that (2n)!!=2^nn!. For n=0, 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)!,
(10)

it follows that

(2n-1)!!=((2n)!)/(2^nn!).
(11)

For n odd,

(n!)/(n!!)=(n(n-1)(n-2)...(1))/(n(n-2)(n-4)...(1))
(12)
=(n-1)(n-3)...(1)
(13)
=(n-1)!!.
(14)

For n even,

(n!)/(n!!)=(n(n-1)(n-2)...(2))/(n(n-2)(n-4)...(2))
(15)
=(n-1)(n-3)...(2)
(16)
=(n-1)!!.
(17)

Therefore, for any n,

(n!)/(n!!)=(n-1)!!
(18)
n!=n!!(n-1)!!.
(19)

The double factorial satisfies the beautiful series

sum_(n=0)^(infty)(x^(2n))/((2n)!!)=e^(x^2/2)
(20)
sum_(n=0)^(infty)(x^(2n+1))/((2n+1)!!)=sqrt(pi/2)erf(x/(sqrt(2)))e^(x^2/2)
(21)
sum_(n=0)^(infty)(x^n)/(n!!)=1/2e^(x^2/2)[sqrt(2pi)erf(x/(sqrt(2)))+2].
(22)

The latter gives rhe sum of reciprocal double factorials in closed form as

sum_(n=0)^(infty)1/(n!!)=sqrt(e)[1+sqrt(pi/2)erf(1/2sqrt(2))]
(23)
=sqrt(e)[1/2sqrt(2)+gamma(1/2,1/2)]
(24)
=3.0594074053425761445...
(25)

(OEIS A143280), where gamma(a,x) 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

sum_(n=0)^infty(-1)^n[((2n-1)!!)/((2n)!!)]^3=[(Gamma(9/8))/(Gamma(5/4)Gamma(7/8))]^2
(26)

(Hardy 1999, p. 106). Whipple (1926) gives a generalization of this sum (Hardy 1999, pp. 111-112).

See also

Barnes G-Function, Double Factorial Prime, Factorial, Gamma Function, Multifactorial

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/Factorial2/

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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.

Referenced on Wolfram|Alpha

Double Factorial

Cite this as:

Weisstein, Eric W. "Double Factorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DoubleFactorial.html

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