Introductions to Factorial Introduction to the factorials and binomials General The factorials and binomials have a very long history connected with their natural appearance in combinatorial problems. Such combinatorial-type problems were known and partially solved even in ancient times. The first mathematical descriptions of binomial coefficients arising from expansions of a + b n for n 2, 3, 4, ¼ appeared in the works of Chia Hsien (1050), al-Karaji (about 1100), Omar al-Khayyami (1080), Bhaskara Acharya (1150), al- Samaw'al (1175), Yang Hui (1261), Tshu shi Kih (1303), Shih–Chieh Chu (1303), M. Stifel (1544), Cardano (1545), Scheubel (1545), Peletier (1549), Tartaglia (1556), Cardan (1570), StevinH (1585),L Faulhaber (1615), Girard (1629), Oughtred (1631), Briggs (1633), Mersenne (1636), Fermat (1636), Wallis (1656), Montmort (1708), and De Moivre (1730). B. Pascal (1653) gave a recursion relation for the binomial, and I. Newton (1676) studied its cases with fractional arguments. It was known that the factorial n! grows very fast. Its growth speed was estimated by J. Stirling (1730) who found the famous asymptotic formula for the factorial named after him. A special role in the history of the factorial and binomial belongs to L. Euler, who introduced the gamma function G z as the natural extension of factorial (n != G n + 1 ) for noninteger arguments and used notations with parentheses for the binomials (1774, 1781). C. F. Hindenburg (1779) used not only binomials but introduced multinomials as their generalizations. The modern H L notation n! was suggested by C. Kramp (1808, 1816). C. F. Gauss (1812) also widely used binomials in his mathe- H L n matical research, but the modern binomial symbol k was introduced by A. von Ettinghausen (1826); later Förstemann (1835) gave the combinatorial interpretation of the binomial coefficients. A. L. Crelle (1831) used a symbol that notates the generalizedJ N factorial a a + 1 a + 2 ¼ a + n - 1 . Later P. E. Appell (1880) ascribed the name Pochhammer symbol for the notation of this product because it was widely used in the research of L. A. Pochhammer (1890). H L H L H L While the double factorial n!! was introduced long ago, its extension for complex arguments was suggested only several years ago by J. Keiper and O. I. Marichev (1994) during the implementation of the function Factorial2 in Mathematica. The classical combinatorial applications of the factorial and binomial functions are the following: è The factorial n! gives the number of possible placements of n people on n chairs. n è The binomial k gives the number of possible selections of k numbers from a larger group of n numbers, for instance on a lotto strip. J N http://functions.wolfram.com 2 è The multinomial n; n1, n2, ¼, nm is the number of ways of putting n n1 + n2 + ¼ nm different objects into th m different boxes with nk in the k box, k 1, 2, ¼, m. H L Definitions of factorials and binomials n The factorial n!, double factorial n!!, Pochhammer symbol a n, binomial coefficient k , and multinomial coeffi- cient n1 + n2 + ¼ + nm; n1, n2, ¼, nm are defined by the following formulas. The first formula is a general definition for the complex arguments, and the second one is HforL positive integer arguments:J N n! GH n + 1 L n n! k ; n Î N+ H L k=1 1 1-cos p n ä2 4 n n!! 2n 2 G + 1 p 2 H H LL G a + n a n ; Ø -a ÎK Z -Oa ³ 0 n Î Z n £ -a G a n-H1 L H L H H + ß ß ß LL a n Ha L+ k ; n Î N k=0 H nL äH G Ln+ 1 n! ; Ø n Î Z k Î Z k £ n < 0 k G k + 1 G n - k + 1 k! n - k ! n H L 0 ; n Î Z k Î Z k £ n < 0 H H ì ì LL k H L H L H L G n + 1 m n + n +¼ + n ì; n , n , ¼ì , n ; -n Ï N+ n n 1 2 m 1 2 m m k k=1 G nk + 1 k=1 H L m H L + í â n1 + n2 + ¼ + nm; n1, n2, ¼, nm 0Û; -nHÎ N L n nk. k=1 HRemark about values at specialL points: For aí= a andâ n n integers with a £ 0 and n £ -a, the Pochhammer symbol a n cannot be uniquely defined by a limiting procedure based on the previous definition because the two variables a and n can approach the integers a and n with a £ 0 and n £ -a at different speeds. For such integers with a £ 0, n £ -a, the following definition is used: H L -1 n -a ! a n ; -a Î N n Î Z n £ -a. -a - n ! H L H L H L ß ß H L http://functions.wolfram.com 3 n Similarly, for n = n, k k negative integers with k £ n, the binomial coefficient k cannot be uniquely defined by a limiting procedure based on the previous definition because the two variables n, k can approach negative integers n, k with k £ n at different speeds. For negative integers with k £ n, the followingJ definitionN is used: n 0 ; n Î Z k Î Z k £ n < 0. k The previous symbols are interconnected and belong to one group that can be called factorials and binomials. ì ì These symbols are widely used in the coefficients of series expansions for the majority of mathematical functions. A quick look at the factorials and binomials Here is a quick look at the graphics for the factorial the real axis. 7.5 5 x ! 2.5 f 0 x !! -2.5 -5 -7.5 -6 -4 -2 0 2 4 x And here is a quick view of the bivariate binomial and Pochhammer functions. For positive arguments, both functions are free of singularities. For negative arguments, the functions have a complicated structure with many singularities. x x y y i y H L j z k { 5 5 2.5 4 2.5 4 0 0 -2.5 2 -2.5 2 - -5 5 -4 0 -4 0 y y -2 -2 - 0 -2 0 2 x 2 x 2 - - 4 4 4 4 Connections within the group of factorials and binomials and with other function groups Representations through more general functions Two factorials n! and n!! are the particular cases of the incomplete gamma function G a, z with the second argument being 0: n! G n + 1, 0 ; Re n > -1 H L H L H L http://functions.wolfram.com 4 1 1-cos p n 2 4 n n!! 2n 2 G + 1, 0 ; Re n > -2. p 2 H H LL Representations through related equivalent functions K O H L n The factorial n!, double factorial n!!, Pochhammer symbol a n, binomial coefficient k , and multinomial coeffi- cient n1 + n2 + ¼ + nm; n1, n2, ¼, nm can be represented through the gamma function by the following formulas: n! G n + 1 H L J N H L 1 cos p n -1 p 4 n n!! 2Hn 2 L G + 1 2 2 H H L L G a + n a n K O G a -H1 n GL1 - a HaLn ; n Î Z G 1H -L a - n GH aL+ nH L HaLn GH a L -H1 n GL1 - a HaLn ; n Î Z G 1H -L a - n n H L GH n + 1L H L k G Hk + 1 G 1L- k + n H L G n + 1 m n + n + ¼ + n ; n , n , ¼, n ; -n Ï N+ n n . 1 2 H L mH 1 2 L m m k k=1 G nk + 1 k=1 H L HMany of these formulas are usedL as the main elements of theí definitionsâ of many functions. Û H L Representations through other factorials and binomials n The factorials and binomials n!, n!!, a n, k , and n1 + n2 + ¼ + nm; n1, n2, ¼, nm are interconnected by the following formulas: 1 1 H L J N H L cos 2 n p -1 -n sin2 n p n! 2 4 p 2 2 n !! n! n -H 1H !! nL !!L H L H L n! 1 n H L 1 cos p n -1 p 4 n n!! H2Ln 2 ! 2 2 H H L L 1 p cos p n -1 n 2 4 K O n!! 2 1 n 2 2 H H L L H L http://functions.wolfram.com 5 m + n - 1 ! m n ; Ø -m Î N -m - n Î N m - 1 ! H -1 n m!L H L H ß L -m n ; m Î N n Î N Hm - nL ! H a +L k - 1 a + k - 1 Ha k L k! k! ß H kL a - 1 a n n! a - 1 + n; a - 1, n H L n n! H kL nH- k ! k! L n 1 - k + n k k H kL! H L n k + 1 n-k k n - k ! H k L n -1 -n k ; k Î Z k H k!L n H L H L n; n - k, k k n! m n + n H+ ¼ + n ;Ln , n , ¼, n ; n n 1 2 m 1 2 m m k k=1 nk ! k=1 nm + 1 m H L n-nm â n1 + n2 + ¼ + nm; n1, n2, ¼, nm Û ; n nk. m-1 k=1 nk ! k=1 H L TheH best-known propertiesL and formulas âfor factorials and binomials Û Real values for real arguments n For real values of arguments, the values of the factorials and binomials n!, n!!, a , , and n k n1 + n2 + ¼ + nm; n1, n2, ¼, nm are real (or infinity).