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Function defined by a hypergeometric series
the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many
Hypergeometric_function
Discrete probability distribution
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k {\displaystyle
Hypergeometric_distribution
Topics referred to by the same term
Hypergeometric may refer to several distinct concepts within mathematics: The hypergeometric function, a solution to the Gaussian hypergeometric differential
Hypergeometric
Family of power series in mathematics
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function
Generalized hypergeometric function
Generalized_hypergeometric_function
Q-analog of hypergeometric series
mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in
Basic_hypergeometric_series
Solution of a confluent hypergeometric equation
a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential
Confluent hypergeometric function
Confluent_hypergeometric_function
Discrete probability distribution
In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without
Negative hypergeometric distribution
Negative_hypergeometric_distribution
list of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function
List of hypergeometric identities
List_of_hypergeometric_identities
Contour integral involving a product of gamma functions
William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation
Barnes_integral
Special function in mathematics
Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet. The Kampé de Fériet function
Kampé_de_Fériet_function
Equalities involving sums over the coefficients occurring in hypergeometric series
mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These
Hypergeometric_identity
Well defined hypergeometric series discovered by Giuseppe Lauricella
In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are (Lauricella 1893): F A ( 3 ) ( a
Lauricella hypergeometric series
Lauricella_hypergeometric_series
theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities
Fisher's noncentral hypergeometric distribution
Fisher's_noncentral_hypergeometric_distribution
Wallenius' noncentral hypergeometric distribution (named after Kenneth Ted Wallenius) is a generalization of the hypergeometric distribution where items
Wallenius' noncentral hypergeometric distribution
Wallenius'_noncentral_hypergeometric_distribution
Hypergeometric function in mathematics
mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced
General hypergeometric function
General_hypergeometric_function
On finite sums of products of three binomial coefficients, and a hypergeometric sum
sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem
Dixon's_identity
Mathematical series
In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio an/an+1 of two terms is a rational
Bilateral hypergeometric series
Bilateral_hypergeometric_series
Monochrome light beam whose amplitude envelope is a Gaussian function
gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified
Gaussian_beam
Elliptic analog of hypergeometric series
elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series
Elliptic hypergeometric series
Elliptic_hypergeometric_series
following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand
Frobenius solution to the hypergeometric equation
Frobenius_solution_to_the_hypergeometric_equation
Set of four hypergeometric series
four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series
Appell_series
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Hypergeometric distribution
In statistics, the hypergeometric distribution is the discrete probability distribution generated by picking colored balls at random from an urn without
Noncentral hypergeometric distributions
Noncentral_hypergeometric_distributions
Pair of functions in combinatorics
sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent
Wilf–Zeilberger_pair
Polynomial sequence
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are
Jacobi_polynomials
functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ϕ {\displaystyle \phi } by J ν ( 1 ) ( x ; q ) = ( q ν + 1
Jackson_q-Bessel_function
Fast method for calculating the digits of π
{163}}}{2}}\right)=-640320^{3}} , and on the following rapidly convergent generalized hypergeometric series: 1 π = 10005 4270934400 ∑ k = 0 ∞ ( − 1 ) k ( 6 k ) ! ( 545140134
Chudnovsky_algorithm
Types of special mathematical functions
{z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s − 1 z
Incomplete_gamma_function
English mathematician
there) working on hypergeometric functions, who introduced the Hahn–Exton q-Bessel function. Exton, Harold (1976), Multiple hypergeometric functions and applications
Harold_Exton
Nonlinear differential operator used to study conformal mappings
projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions
Schwarzian_derivative
–2t/(1–t2) An explicit expression for them in terms of the generalized hypergeometric function 3F0: s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 −
Mott_polynomials
theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 −
Q-theta_function
Russian mathematician (born 1962)
Kapranov investigated generalized Euler integrals, A {\displaystyle A} -hypergeometric functions, A {\displaystyle A} -discriminants, and hyperdeterminants
Mikhail_Kapranov
Generalization of the hypergeometric differential equation
equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur
Riemann's differential equation
Riemann's_differential_equation
Probability distribution
the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n
Binomial_distribution
Branch of discrete mathematics
function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials & approximations Factorial
Combinatorics
Mathematical theorem on convolved binomial coefficients
Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that 2 F 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c −
Vandermonde's_identity
Bangladeshi Canadian mathematician and writer (1932–2015)
mathematician and writer. He specialized in fields of mathematics such as hypergeometric series and orthogonal polynomials. He also had interests encompassing
Mizan_Rahman
Kummer's function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions
List of mathematical functions
List_of_mathematical_functions
Chinese-American mathematician
American mathematician whose research concerns modular forms, arithmetic hypergeometric functions, as well as number theory in general. She is the Micheal F
Ling_Long_(mathematician)
British mathematician (1922-2008)
Slater (5 January 1922 – 6 June 2008) was a mathematician who worked on hypergeometric functions, and who found many generalizations of the Rogers–Ramanujan
Lucy_Joan_Slater
Type of polynomial sequence
class of Appell polynomials can be obtained in terms of the generalized hypergeometric function. Let Δ ( k , − n ) {\displaystyle \Delta (k,-n)} denote the
Appell_sequence
Special functions used to build correlation functions in 2D CFTs
of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated
Virasoro_conformal_block
Sequence of differential equation solutions
{1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L n ( α ) ( x ) := ( n + α
Laguerre_polynomials
Concept in differential equation mathematics
higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation
Regular_singular_point
Canonical solutions of the general Legendre equation
} is the gamma function and 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function 2 F 1 ( α , β ; γ ; z ) = Γ ( γ ) Γ ( α ) Γ ( β ) ∑ n = 0 ∞
Associated Legendre polynomials
Associated_Legendre_polynomials
Indian mathematician (1887–1920)
another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately
Srinivasa_Ramanujan
Linear recurrence equation
and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions. Let K {\textstyle \mathbb {K} } be a field
P-recursive_equation
Type of mathematical generalization
known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. q-analogs are most
Q-analog
Mathematical family
the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Hahn
Little_q-Jacobi_polynomials
Hungarian-born British mathematician
expert on special functions, particularly orthogonal polynomials and hypergeometric functions. He was born Arthur Diamant in Budapest, Hungary to Ignác
Arthur_Erdélyi
Summation method for hypergeometric terms
Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n)
Gosper's_algorithm
Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized
List of eponyms of special functions
List_of_eponyms_of_special_functions
Mathematical formula by Thomas Clausen
Clausen (1828), expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states 2 F 1 [ a b a + b + 1 / 2 ; x
Clausen's_formula
Game of chance
numbers that are picked on each ticket. Keno probabilities come from a hypergeometric distribution. For Keno, one calculates the probability of hitting exactly
Keno
Israeli mathematician
Rutgers University. Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. He gave the first proof of the alternating
Doron_Zeilberger
Russian number theorist
Russian mathematician and number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and
Wadim_Zudilin
Class of differential equations expressible in differential algebra
the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer
Algebraic differential equation
Algebraic_differential_equation
Statistical significance test
by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table. This setting
Fisher's_exact_test
American mathematician (1906–1996)
mathematician and Catholic religious sister. She is most noted for her work on hypergeometric functions and linear algebra. Fasenmyer grew up in Pennsylvania's oil
Mary_Celine_Fasenmyer
hypergeometric series, dating back to 1837. which cites to Kummer, Ernst Eduard (1836). "Uber die Hypergeometrische Reihe" [About the hypergeometric series]
Perimeter_of_an_ellipse
Classification of orthogonal polynomials
scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials
Askey_scheme
a casino roulette, or the first card of a well-shuffled deck. The hypergeometric distribution, which describes the number of successes in the first m
List of probability distributions
List_of_probability_distributions
Discrete probability distribution
special case where α and β are integers is also known as the negative hypergeometric distribution. The beta distribution is a conjugate distribution of the
Beta-binomial_distribution
Number of subsets of a given size
{\displaystyle \alpha } . Binomial transform Delannoy number Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation
Binomial_coefficient
Russian mathematician (born 1954)
ISBN 978-0-8176-4566-3. Schechtman, V. V.; Varchenko, A. N. (1990). "Hypergeometric solutions of Knizhnik-Zamolodchikov equations". Lett. Math. Phys. 20
Vadim_Schechtman
Technique in information retrieval
Randomness Model is based on the Bernoulli model and its limiting forms, the hypergeometric distribution, Bose–Einstein statistics and its limiting forms, the compound
Divergence-from-randomness model
Divergence-from-randomness_model
Polynomial sequence
Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_polynomials
Jamaican statistician
statistics, zonal polynomials, distance correlation, total positivity, and hypergeometric functions of matrix argument. He is a distinguished professor emeritus
Donald Richards (statistician)
Donald_Richards_(statistician)
Mathematical technique for improving convergence
Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Given an infinite series
Series_acceleration
Multivalued function in mathematics
stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as V = V 0 1 + W ( e − x σ ) . {\displaystyle
Lambert_W_function
Probability distribution
the plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1 F 1 {\textstyle {}_{1}F_{1}} and U . {\textstyle U.} E
Normal_distribution
Infinite sum
{z^{n}}{n!}}} and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and
Series_(mathematics)
German mathematician (1821–1881)
functions (Handbuch der Kugelfunctionen). He also investigated basic hypergeometric series. He introduced the Mehler–Heine formula. Heinrich Eduard Heine
Eduard_Heine
Generalization of the hypergeometric function
particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's
Meijer_G-function
Family of orthogonal polynomials
polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 (Chebyshev
Hahn_polynomials
Input to a mathematical function
{\displaystyle y} , in an ordered pair ( x , y ) {\displaystyle (x,y)} . The hypergeometric function is an example of a four-argument function. The number of arguments
Argument_of_a_function
Generating pseudo-random numbers that follow a probability distribution
Exponential F Gamma Geometric Gumbel Hypergeometric Laplace Logistic Log-normal Logarithmic Multinomial Multivariate hypergeometric Multivariate normal Negative
Non-uniform random variate generation
Non-uniform_random_variate_generation
mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter
Continuous q-Laguerre polynomials
Continuous_q-Laguerre_polynomials
Uses of the constant
{\displaystyle n\to \infty } . With 2 F 1 {\displaystyle {}_{2}F_{1}} being the hypergeometric function: ∑ n = 0 ∞ r 2 ( n ) q n = 2 F 1 ( 1 2 , 1 2 , 1 , z ) {\displaystyle
List_of_formulae_involving_π
Science of classifying organisms
phylogeny or evolutionary relationships. It results in a measure of hypergeometric "distance" between taxa. Phenetic methods have become relatively rare
Taxonomy_(biology)
Algorithmic technique
series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. Given a series S ( a , b ) = ∑ n = a b p
Binary_splitting
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Real root of the polynomial x^5+x+a
ordinary differential equation of hypergeometric type, whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's
Bring_radical
French mathematician (1858–1936)
Goursat also published texts on partial differential equations and hypergeometric series. Edouard Goursat was born in Lanzac, Lot. He was a graduate of
Édouard_Goursat
British mathematician
1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several q-analogs such as the Jackson–Bessel functions
F._H._Jackson
Mathematical functions
increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun,
Falling_and_rising_factorials
orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q
Big_q-Legendre_polynomials
Computer program
Shigeru (2011). "10 trillion digits of pi: A case study of summing hypergeometric series to high precision on multicore systems" (PDF). Alexander Jih-Hing
Y-cruncher
Concept in probability theory and statistics
multivariate normal, other elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial, or Dirichlet distribution, but not
Partial_correlation
}&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson
Askey–Wilson_polynomials
Mental exercise in probability and statistics
number of draws before the first successful (correctly colored) draw. hypergeometric distribution: the balls are not returned to the urn once extracted.
Urn_problem
Mathematical identities related to integer partitions
the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered
Rogers–Ramanujan_identities
Mathematical function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function
Beta_function
Concept in combinatorics (part of mathematics)
theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike
Q-Pochhammer_symbol
Algorithmic runtime requirements for common math procedures
O{\mathord {\left(M(n)n^{1/2}(\log n)^{2}\right)}}} Fixed rational number Hypergeometric series O ( M ( n ) ( log n ) 2 ) {\displaystyle O{\mathord {\left(M(n)(\log
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Mathematical function for the probability a given outcome occurs in an experiment
hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution
Probability_distribution
In physics, solution to Schrödinger equation
particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb
Coulomb_wave_function
German mathematician (1826–1866)
mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces. In the field of real
Bernhard_Riemann
American mathematician (1933–2019)
which organizes orthogonal polynomials of ( q {\displaystyle q} -)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials
Richard_Askey
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
Surname or Lastname
English
English : habitational name from any of the numerous minor places so called, from Old English hol ‘hollow’, ‘sunken’ + weg ‘way’, ‘path’. In Ireland, it has sometimes been Gaelicized as Ó hAilmhic (see Hulvey).
Boy/Male
Irish
Name of a saint.
Girl/Female
Australian, British, Danish, English, French, German, Greek
Form of Eleanor; Light
Girl/Female
Indian
Surname or Lastname
English
English : topographic name from Middle English whit ‘white’ + halgh ‘nook’ or hall ‘hall’.
Girl/Female
Hindu, Indian
Golden Body
Boy/Male
Muslim
Religion
Boy/Male
Indian
Full of qualities, Expansionist, Vast, Spacious, Man of qualities
Female
Icelandic
Icelandic form of Old Norse Ragnhildr, RAGNHILDUR means "battle counsel."
Boy/Male
Hindu
Lord Shiva, The cosmic serpent Shesh
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC