\end{alignedat}}} Falling and rising factorials of integers are directly related to the ordinary factorial: n ! = 1 ( n ) = ( n ) n , ( m...
18 KB (3,300 words) - 00:11, 1 September 2024
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...
28 KB (4,028 words) - 11:56, 1 November 2024
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 (1,821 words) - 22:59, 8 November 2024
coefficients) of factorials j ! {\displaystyle j!} ; in particular P {\displaystyle P} is an integer. Smaller factorials divide larger factorials, so the smallest...
51 KB (6,747 words) - 02:30, 12 December 2024
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) - 22:05, 9 April 2024
notation Metric tensor Musical isomorphism Falling and rising factorials Dual space § Bilinear products and dual spaces Kay, D. C. (1988). Tensor Calculus...
16 KB (3,715 words) - 20:40, 22 December 2024
Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof...
2 KB (218 words) - 16:34, 12 March 2022
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
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) - 05:39, 4 July 2024
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 (6,914 words) - 09:18, 30 November 2024
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
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) - 01:59, 4 September 2023
Concrete Mathematics (section Contents and history)
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
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 (10,762 words) - 17:32, 20 December 2024
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...
24 KB (4,054 words) - 17:48, 20 December 2024
symbol used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial. This series is also known...
10 KB (1,614 words) - 04:22, 3 December 2024
600 (number) (redirect from Six hundred and four)
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 (3,989 words) - 16:38, 30 November 2024
Chinese restaurant process (section Relationship between Dirichlet-categorical and one-parameter CRP)
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
73 (number) (category CS1 maint: date and year)
These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number...
21 KB (2,902 words) - 12:36, 21 December 2024
subjective observations by workers falling from scaffolds, soldiers who suffered injuries, climbers who had fallen from heights and other individuals who had come...
72 KB (8,379 words) - 04:18, 21 December 2024
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 (1,316 words) - 19:50, 18 December 2024
Regge theory (section History and implications)
{\displaystyle \Gamma (x)} is the gamma function, a generalization of factorial ( x − 1 ) ! {\displaystyle (x-1)!} . This gamma function is a meromorphic...
13 KB (1,923 words) - 04:36, 30 November 2024
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...
71 KB (10,620 words) - 17:42, 9 December 2024
Suicide (redirect from Suicide and evolution)
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"...
178 KB (18,448 words) - 02:36, 7 December 2024
mathematics and logic that defines a function based on the values it takes on smaller arguments, essential for defining functions like factorials and other...
270 KB (30,192 words) - 23:49, 3 December 2024
Generating function transformation (section Example: A double factorial integral for the EGF of the Stirling numbers of the second kind)
generating functions for generalized factorial functions formed as special cases of the generalized rising factorial product functions, or Pochhammer k-symbol...
62 KB (11,140 words) - 18:07, 13 October 2024
Gurmukhi (section History and development)
although Punjabi lacks these sounds. Tones in Punjabi can be either rising, neutral, or falling: When the tonal letter is in onset positions, as in the pronunciation...
72 KB (5,715 words) - 03:50, 23 December 2024
Inuit phonology (redirect from Inuktitut Phonology and Phonetics)
followed by a falling pitch on the second syllable means "What did you say?" A middle pitch on the first syllable followed by a rising pitch on the second...
22 KB (2,017 words) - 06:11, 26 August 2024
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 (7,259 words) - 18:02, 20 December 2024