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double factorial

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double factorial

Etymology

Fromdouble +‎factorial.

Noun

double factorial (pluraldouble factorials)

(mathematics, combinatorics) For an odd integer, the result of multiplying all the odd integers from 1 to the given number; or for an even integer, the result of multiplying all the even integers from 2 to the given number; symbolised by a double exclamation mark (!!). For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.
Synonym:(rare)semifactorial
- 1902, Arthur Schuster, “On some Definite Integrals and a New Method of reducing a Function of Spherical Co-ordinates to a Series of Spherical Harmonics”, inProceedings of the Royal Society of London, volume71,→DOI,→JSTOR, page99:
  The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, $n\cdot n-2\cdot n-4\cdots 1$ , if $n {\displaystyle n}$ be odd, or $n\cdot n-2\cdots 2$ if $n {\displaystyle n}$ be odd^{[sic – meaningeven]}. I propose to write $n!! {\displaystyle n!!}$ for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial." Full advantage of the new symbol is only gained by extending its meaning to the negative values of $n {\displaystyle n}$ . Its complete definition may then be included in the equations $n!!=n(n-2)!!,\quad 1!!=1,\quad 2!!=2.$
- 1948 September, B. E. Meserve, “Double Factorials”, inThe American Mathematical Monthly, volume55, number 7,→DOI,→JSTOR, page425:
  Thedouble factorial notation ${\begin{aligned}(2n)!!&=2\cdot 4\cdot 6\cdots (2n-2)(2n)=2^{n}n!\\(2n+1)!!&=1\cdot 3\cdot 5\cdots (2n-1)(2n+1)={\frac {(2n+1)!}{2^{n}n!}}\end{aligned}}$ may be considered as a generalization of $n!=1\cdot 2\cdot 3\cdots n$ .
- 1958–1959, Kenneth W. Ford, E. J. Konopinski, “Evaluation of Slater integrals with harmonic oscillator wave functions”, inNuclear Physics, volume 9, number 2,→DOI, page219:
  We prefer now to write the expansion in a slightly different way in order to exhibit more clearly the symmetry properties of the expansion coefficients: $f_{k}(m,m')=({\tfrac {1}{2}}\pi )^{\tfrac {1}{2}}2^{-2{\bar {m}}-3}\sum _{\sigma }(2{\bar {m}}-2\sigma +1)!!\,T_{k}^{\,2\sigma }(m,m')J_{2\sigma },$ where, as above, ${\bar {m}}$ is the average of $m {\displaystyle m}$ and $m^{'} {\displaystyle m'}$ , ${\bar {m}}={\tfrac {1}{2}}(m+m')$ , and thedouble factorial notation is used, $(2n+1)!!=1\cdot 3\cdot 5\cdots (2n+1)$ .
- 2008 April, Adriana Pálffy, Jörg Evers, Christoph H. Keitel, “Electric-dipole-forbidden nuclear transitions driven by super-intense laser fields”, inPhysical Review C, volume77, number 4,→DOI, page044602-3:
  The symbol $!! {\displaystyle !!}$ in Eq. (9) denotes thedouble factorial given by $n!!=n(n-2)(n-4)\dots \kappa _{n}$ , where $\kappa _{n}$ is $1 {\displaystyle 1}$ for odd $n {\displaystyle n}$ and $2 {\displaystyle 2}$ for even $n {\displaystyle n}$ .
- 2012 June, Henry Gould, Jocelyn Quaintance, “Double Fun with Double Factorials”, inMathematics Magazine, volume85, number 3,→DOI, pages177–178:
  Double factorials can also be defined recursively. Just as we can define the ordinary factorial by $n!=n\cdot (n-1)!$ for $n\geq 1$ with $0!=1$ , we can define thedouble factorial by $n!!=n\cdot (n-2)!!$ for $n\geq 2$ with initial values $0!!=1!!=1$ . With our convention that $(-1)!!=1$ , the recursion is valid for all positive integers $n {\displaystyle n}$ .

Translations

mathematics

Further reading

MathWorld,Double Factorial

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