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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
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
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
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
list of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function
List of hypergeometric identities
List_of_hypergeometric_identities
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
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
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
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
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
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
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
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
Polynomial sequence
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are
Jacobi_polynomials
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
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
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
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
Branch of discrete mathematics
function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials & approximations Factorial
Combinatorics
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
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
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
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
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
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
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
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
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
–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
Family of basic hypergeometric orthogonal polynomials
mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter
Continuous big q-Hermite polynomials
Continuous_big_q-Hermite_polynomials
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
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
Mathematics
polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by S n ( x 2 ; a
Continuous dual Hahn polynomials
Continuous_dual_Hahn_polynomials
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
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
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
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
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
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
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
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
1888. F. J. Whipple coined the phrase "Saalschützian" for generalized hypergeometric 3 F 2 {\displaystyle _{3}F_{2}} series where one of the numerator parameters
Louis_Saalschütz
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
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
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
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
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
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)
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
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)
Family of hypergeometric polynomials
In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by Mary Celine Fasenmyer in 1947. They include Legendre
Sister_Celine's_polynomials
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
{\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0} , (f periodic) Physics Hypergeometric 2 z ( 1 − z ) d 2 w d z 2 + [ c − ( a + b + 1 ) z ] d w d z − a b w
List of linear ordinary differential equations
List_of_linear_ordinary_differential_equations
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
Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers. Define μ {\displaystyle
Ferrers_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
}&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
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
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
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
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
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
_{1}(0;q^{\nu +1};q,qx^{2}).} ϕ {\displaystyle \phi } is the basic hypergeometric function. Koelink and Swarttouw proved that J ν ( 3 ) ( x ; q ) {\displaystyle
Hahn–Exton_q-Bessel_function
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
Venezuelan-American mathematician
in 2009 at the University of Texas at Austin, with the dissertation Hypergeometric Functions in Arithmetic Geometry supervised by Fernando Rodríguez-Villegas
Adriana_Salerno
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
British mathematician
introduced Bailey's lemma and Bailey pairs into the theory of basic hypergeometric series. Bailey chains and Bailey transforms are named after him. Slater
Wilfrid_Norman_Bailey
American mathematician (1916–1994)
Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series. He is best known for his lecture notes on the subject which
Nathan_Fine
Concept in probability theory and statistics
multivariate normal, other elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial, or Dirichlet distribution, but not
Partial_correlation
Generalization of the Meijer G-function and the Fox–Wright function
Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. (Srivastava & Manocha
Fox_H-function
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
South african mathematician
University of the Witwatersrand. Her 2001 dissertation, Zeros of general hypergeometric polynomials, was supervised by Kathy Driver. She was a member of the
Kerstin_Jordaan
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
American mathematician
polynomials and basic hypergeometric series, who introduced the Askey–Gasper inequality. Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia
George_Gasper
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
Japanese mathematician
Aomoto is a Japanese mathematician who introduced the Aomoto-Gel'fand hypergeometric function and the Aomoto integral. He was a professor at Nagoya University
Kazuhiko_Aomoto
In mathematics, a solution to a modified form of the confluent hypergeometric equation
solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving
Whittaker_function
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
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
Probability distribution
characteristic function of the beta distribution is Kummer's confluent hypergeometric function (of the first kind): φ X ( α ; β ; t ) = E [ e i t X ] =
Beta_distribution
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
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
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
Surname or Lastname
English
English : unexplained.possibly an altered form of German Stenger.
Surname or Lastname
English
English : patronymic from the personal name Dick.
Girl/Female
Tamil
Diksheeka | தீகà¯à®·à®¿à®•à®¾
Very silent & simple
Boy/Male
Tamil
Beloved
Boy/Male
Arabic, Muslim
Name of a Pre-islamic Arabic King
Boy/Male
Hebrew
Fertile.
Biblical
same as Ashur
Male
Japanese
(æ£ç”·) Japanese name MASAO means "correct man."
Boy/Male
Indian
One who distinguishes truth from falsehood, Power of discrimination
Girl/Female
Arabic, Farsi, Iranian, Muslim, Parsi
Best Coquetry
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC
HYPERGEOMETRIC