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Double Factorial

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The double factorial of a nonnegative 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 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.

For a nonnegative integer n, the double factorial is related to the gamma function by

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

(Arfken 1985, p. 548). This identity for odd integer arguments of the double factorial is distinct from its extension to complex numbers illustrated below.

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).

DoubleFactorialReIm
DoubleFactorialContours

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
(6)
=[(2n+1)(2n-1)...1][2n(2n-2)(2n-4)...2]
(7)
=(2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4)...2·1
(8)
=(2n+1)!,
(9)

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
(10)
=[2(n)][2(n-1)][2(n-2)]...2
(11)
=2^nn!,
(12)

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)
(13)
=(2n-1)(2n-3)...1·2n(2n-2)(2n-4)...2
(14)
=2n(2n-1)(2n-2)(2n-3)(2n-4)...2(1)
(15)
=(2n)!,
(16)

it follows that

(2n-1)!!=((2n)!)/(2^nn!).
(17)

For a positive odd integer n,

(n!)/(n!!)=(n(n-1)(n-2)...(1))/(n(n-2)(n-4)...(1))
(18)
=(n-1)(n-3)...2
(19)
=(n-1)!!.
(20)

For a positive even integer n,

(n!)/(n!!)=(n(n-1)(n-2)...1)/(n(n-2)(n-4)...2)
(21)
=(n-1)(n-3)...1
(22)
=(n-1)!!.
(23)

Therefore, for any nonnegative integer n,

(n!)/(n!!)=(n-1)!!
(24)
n!=n!!(n-1)!!.
(25)

The double factorial satisfies the beautiful series

sum_(n=0)^(infty)(x^(2n))/((2n)!!)=e^(x^2/2)
(26)
sum_(n=0)^(infty)(x^(2n+1))/((2n+1)!!)=sqrt(pi/2)erf(x/(sqrt(2)))e^(x^2/2)
(27)
sum_(n=0)^(infty)(x^n)/(n!!)=1/2e^(x^2/2)[sqrt(2pi)erf(x/(sqrt(2)))+2].
(28)

The latter gives the 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))]
(29)
=sqrt(e)[1+(gamma(1/2,1/2))/(sqrt(2))]
(30)
=3.0594074053425761445...
(31)

(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
(32)

(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

https://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 Resource. https://mathworld.wolfram.com/DoubleFactorial.html

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