double factorial

double +‎ factorial

double factorial (plural double factorials)

1. (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”, in Proceedings of the Royal Society of London, volume 71, →DOI, →JSTOR, page 99:

     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, ${\displaystyle n\cdot n-2\cdot n-4\cdots 1}$, if ${\displaystyle n}$ be odd, or ${\displaystyle n\cdot n-2\cdots 2}$ if ${\displaystyle n}$ be odd^{[sic – meaning even]}. I propose to write ${\displaystyle n!!}$ 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 ${\displaystyle n}$. Its complete definition may then be included in the equations

     $${\displaystyle n!!=n(n-2)!!,\quad 1!!=1,\quad 2!!=2.}$$

     

   • 1948 September, B. E. Meserve, “Double Factorials”, in The American Mathematical Monthly, volume 55, number 7, →DOI, →JSTOR, page 425:

     The double factorial notation

     $${\displaystyle {\begin{aligned}(2n)!!&=2\cdot 4\cdot 6\cdots (2n-2)(2n)=2^{n}n!\\(2n+1)!!&=1\cdot 3\cdot 5\cdots (2n-1)(2n+1)={\frac {(2n+1)!}{2^{n}n!}}\end{aligned}}}$$

     

     may be considered as a generalization of ${\displaystyle n!=1\cdot 2\cdot 3\cdots n}$.

   • 1958–1959, Kenneth W. Ford, E. J. Konopinski, “Evaluation of Slater integrals with harmonic oscillator wave functions”, in Nuclear Physics, volume 9, number 2, →DOI, page 219:

     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:

     $${\displaystyle f_{k}(m,m')=({\tfrac {1}{2}}\pi )^{\tfrac {1}{2}}2^{-2{\bar {m}}-3}\sum _{\sigma }(2{\bar {m}}-2\sigma +1)!!\,T_{k}^{\,2\sigma }(m,m')J_{2\sigma },}$$

     

     where, as above, ${\displaystyle {\bar {m}}}$ is the average of ${\displaystyle m}$ and ${\displaystyle m'}$, ${\displaystyle {\bar {m}}={\tfrac {1}{2}}(m+m')}$, and the double factorial notation is used, ${\displaystyle (2n+1)!!=1\cdot 3\cdot 5\cdots (2n+1)}$.

   • 2008 April, Adriana Pálffy, Jörg Evers, Christoph H. Keitel, “Electric-dipole-forbidden nuclear transitions driven by super-intense laser fields”, in Physical Review C, volume 77, number 4, →DOI, page 044602-3:

     The symbol ${\displaystyle$ !!} in Eq. (9) denotes the double factorial given by ${\displaystyle n!!=n(n-2)(n-4)\dots \kappa _{n}}$, where ${\displaystyle \kappa _{n}}$ is ${\displaystyle 1}$ for odd ${\displaystyle n}$ and ${\displaystyle 2}$ for even ${\displaystyle n}$.

   • 2012 June, Henry Gould, Jocelyn Quaintance, “Double Fun with Double Factorials”, in Mathematics Magazine, volume 85, number 3, →DOI, pages 177–178:

     Double factorials can also be defined recursively. Just as we can define the ordinary factorial by ${\displaystyle n!=n\cdot (n-1)!}$ for ${\displaystyle n\geq 1}$ with ${\displaystyle 0!=1}$, we can define the double factorial by

     $${\displaystyle n!!=n\cdot (n-2)!!}$$

     

     for ${\displaystyle n\geq 2}$ with initial values ${\displaystyle 0!!=1!!=1}$. With our convention that ${\displaystyle (-1)!!=1}$, the recursion is valid for all positive integers ${\displaystyle n}$.

• MathWorld, Double Factorial