• \end{alignedat}}} Falling and rising factorials of integers are directly related to the ordinary factorial: n ! = 1 ( n ) = ( n ) n , ( m...
    19 KB (3,377 words) - 01:53, 5 April 2025
  • The notation ( x ) n {\displaystyle (x)_{n}} is used to denote the falling factorial, an n-th degree polynomial defined by ( x ) n = x ( x − 1 ) ( x −...
    13 KB (1,824 words) - 08:05, 1 June 2025
  • Stirling number (category Factorial and binomial topics)
    in expansions of falling and rising factorials (also known as the Pochhammer symbol) as polynomials. That is, the falling factorial, defined as   ( x...
    29 KB (4,065 words) - 08:57, 3 June 2025
  • coefficients) of factorials j ! {\displaystyle j!} ; in particular P {\displaystyle P} is an integer. Smaller factorials divide larger factorials, so the smallest...
    52 KB (6,818 words) - 21:52, 1 July 2025
  • Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof...
    2 KB (220 words) - 05:14, 5 March 2025
  • Pochhammer k-symbol (category Factorial and binomial topics)
    special cases of the falling and rising factorials, including the Pochhammer symbol, and the generalized cases of the multiple factorial functions (multifactorial...
    8 KB (1,732 words) - 00:26, 23 May 2025
  • Sheffer sequence (category Factorial and binomial topics)
    polynomials; The Bernoulli polynomials of the second kind; The Falling and rising factorials; The Touchard polynomials; The Mittag-Leffler polynomials; Rota...
    7 KB (1,049 words) - 19:28, 20 June 2025
  • Thumbnail for Hypergeometric function
    Hypergeometric function (category Factorial and binomial topics)
    used in Falling and rising factorials. Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications...
    38 KB (6,920 words) - 21:17, 14 April 2025
  • Generalized Pochhammer symbol q-Pochhammer symbol Pochhammer contour Falling and rising factorials Works by or about Leo August Pochhammer at the Internet Archive...
    3 KB (348 words) - 16:21, 9 December 2024
  • Thumbnail for Lah number
    Lah number (category Factorial and binomial topics)
    mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were...
    11 KB (1,812 words) - 15:40, 30 October 2024
  • Polynomial sequence (category Sequences and series)
    Hermite polynomials Many are studied in algebra and combinatorics: Monomials Rising factorials Falling factorials All-one polynomials Abel polynomials Bell...
    2 KB (176 words) - 15:36, 14 August 2021
  • mathematical notation: the Iverson bracket, floor and ceiling functions, and notation for rising and falling factorials. Donald Knuth used the first edition of...
    6 KB (470 words) - 20:33, 28 November 2024
  • Thumbnail for Stirling numbers of the second kind
    Stirling numbers of the second kind (category Factorial and binomial topics)
    that combinatorialists use for falling factorials coincides with the notation used in special functions for rising factorials; see Pochhammer symbol. Transformation...
    25 KB (4,328 words) - 18:26, 20 April 2025
  • These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number...
    14 KB (2,004 words) - 14:56, 9 April 2025
  • Thumbnail for Binomial coefficient
    Binomial coefficient (category Factorial and binomial topics)
    object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are...
    62 KB (10,790 words) - 13:49, 8 July 2025
  • notation to represent the rising factorial.) Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function...
    3 KB (396 words) - 21:10, 14 April 2025
  • Thumbnail for Near-death experience
    subjective observations by workers falling from scaffolds, soldiers who suffered injuries, climbers who had fallen from heights and other individuals who had come...
    87 KB (9,811 words) - 23:07, 3 July 2025
  • an n X n board, with 1 in each row and column. Also number of permutations of length n without rising or falling successions)". The On-Line Encyclopedia...
    24 KB (3,965 words) - 13:08, 27 June 2025
  • Thumbnail for Suicide
    Toland, The Rising Sun: The Decline and Fall of the Japanese Empire 1936–1945, Random House, 1970, p. 519 O'Keeffe TM (1984). "Suicide and Self-Starvation"...
    182 KB (18,746 words) - 11:50, 3 July 2025
  • fewer for the third element, and so on. Therefore instead of by an ordinary power of x, the value is given by a falling factorial power of x, in which each...
    43 KB (5,609 words) - 19:20, 19 January 2025
  • Stirling numbers of the first kind (category Factorial and binomial topics)
    recurrence relation using the definition of Stirling numbers in terms of rising factorials. Distributing the last term of the product, we have x n + 1 ¯ = x...
    38 KB (7,265 words) - 04:38, 9 June 2025
  • is the rising factorial and x m _ = ∏ i = 0 m − 1 ( x − i ) {\displaystyle x^{\underline {m}}=\prod _{i=0}^{m-1}(x-i)} is the falling factorial. It is...
    24 KB (3,990 words) - 10:53, 6 December 2024
  • Thumbnail for Flynn effect
    Flynn effect (category Race and intelligence controversy)
    draftees in NATO countries in Europe—report raw scores, and those also confirm a trend of rising scores over time. The average rate of increase seems to...
    64 KB (7,695 words) - 13:53, 9 July 2025
  • Thumbnail for Scatter plot
    Scatter plot (category Statistical charts and diagrams)
    example, weight and height would be on the y-axis, and height would be on the x-axis. Correlations may be positive (rising), negative (falling), or null (uncorrelated)...
    12 KB (1,332 words) - 13:51, 9 June 2025
  • Thumbnail for Riemann zeta function
    Riemann zeta function (category Zeta and L-functions)
    1; that context gives rise to a series expansion in terms of the falling factorial. On the basis of Weierstrass's factorization theorem, Hadamard gave...
    74 KB (10,718 words) - 01:21, 7 July 2025
  • is a multi‐factorial event that involves the complex interaction of a number of risk factors. The effect of each individual factor and its importance...
    77 KB (7,316 words) - 14:21, 9 July 2025
  • Multiset (category Factorial and binomial topics)
    \over k!},} to match the expression of binomial coefficients using a falling factorial power: ( n k ) = n k _ k ! . {\displaystyle {n \choose k}={n^{\underline...
    35 KB (4,983 words) - 20:48, 3 July 2025
  • Binomial transform (category Factorial and binomial topics)
    rising k-binomial transform is sometimes defined as ∑ j = 0 n ( n j ) j k a j . {\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}j^{k}a_{j}.} The falling...
    14 KB (2,438 words) - 15:02, 19 April 2025
  • Thumbnail for Vocational education
    qualifications demanded for job entry are rising. This reflects a need for not just a more knowledgeable and skilled workforce, but one that can adapt...
    116 KB (14,288 words) - 14:29, 9 July 2025
  • {\displaystyle \Gamma (x)} is the gamma function, a generalization of factorial ( x − 1 ) ! {\displaystyle (x-1)!} . This gamma function is a meromorphic...
    13 KB (1,919 words) - 06:29, 23 February 2025