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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
Function defined by a hypergeometric series
ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as
Hypergeometric_function
Family of power series in mathematics
generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n {\displaystyle n} is a rational function of n
Generalized hypergeometric function
Generalized_hypergeometric_function
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
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
Q-analog of hypergeometric series
by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the
Basic_hypergeometric_series
Mathematical theorem on convolved binomial coefficients
case of Gauss's hypergeometric theorem, which states that 2F1(a,b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) where 2F1 is the hypergeometric function and Γ(n+1)=n! is
Vandermonde's_identity
Discrete probability distribution
random variable X {\displaystyle X} follows the hypergeometric distribution if its probability mass function (pmf) is given by p X ( k ) = Pr ( X = k ) =
Hypergeometric_distribution
Solutions of Legendre's differential equation
which can both be expressed in terms of the hypergeometric function, 2F1. With Γ being the gamma function, the first solution is Pλμ(z)=1Γ(1−μ)[z+1z−1]μ/22F1(−λ
Legendre_function
of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists
List of hypergeometric identities
List_of_hypergeometric_identities
Contour integral involving a product of gamma functions
product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The
Barnes_integral
Generalization of the hypergeometric function
of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as
Meijer_G-function
In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case
MacRobert_E_function
Type of mathematical series
two consecutive terms is a rational function of n {\displaystyle n} . The definition of the generalized hypergeometric series is similar, except that the
Bilateral hypergeometric series
Bilateral_hypergeometric_series
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 s e
Incomplete_gamma_function
Mathematical equation
the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of
Picard–Fuchs_equation
Analytic function that does not satisfy a polynomial equation
generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values. Transcendental functions cannot
Transcendental_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
special functions, Schwarz's list or the Schwarz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be
Schwarz's_list
Linear recurrence equation
\mathbb {N} }} ). The other algorithms for finding more general solutions (e.g. rational or hypergeometric solutions) also rely on algorithms which compute polynomial
P-recursive_equation
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
Sigmoid shape special function
Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ( x ) = 2 x π M ( 1 2 , 3 2 , − x 2 ) . {\displaystyle
Error_function
Extension of the factorial function
functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented
Gamma_function
German polymath and scholar (1777–1855)
"Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general hypergeometric function F(α,β,γ,x),
Carl_Friedrich_Gauss
Special mathematical functions defined on the surface of a sphere
group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2)
Spherical_harmonics
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_polynomials
Multivalued function in mathematics
solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional
Lambert_W_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward
Sigmoid_function
Mathematical formula involving a given set of operations
functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions are
Closed-form_expression
Concept in probability theory and statistics
to more general cases. The moment generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are
Moment_generating_function
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
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
Function for Heun's differential equation
symmetries of the hypergeometric differential equations obtained by Kummer.[citation needed] The symmetries fixing the local Heun function form a group of
Heun_function
Special function defined by an integral
connexion with the confluent hypergeometric functions is that E 1 {\displaystyle E_{1}} is an exponential times the function U ( 1 , 1 , z ) {\displaystyle
Exponential_integral
Polynomial sequence
Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite: C n ( α
Gegenbauer_polynomials
Generalization of the hypergeometric differential equation
The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is P
Riemann's differential equation
Riemann's_differential_equation
Family of orthogonal polynomials
polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by Qn(x;α,β,N)=3F2(−n,−x,n+α+β+1;α+1,−N+1;1) for n=0,1,…,N. If
Hahn_polynomials
Mathematical function
q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the
Ramanujan_theta_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
Cunningham (1908). It can be defined in terms of the confluent hypergeometric function U, by ω m , n ( x ) = e − x + π i ( m / 2 − n ) Γ ( 1 + n − m /
Cunningham_function
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
Orthogonal polynomials
Carl Charlier in 1905. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 )
Charlier_polynomials
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
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
Type of function in mathematics
special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The gamma function away
Analytic_function
Mathematics concept
version of the hypergeometric differential equation Curiously, they have been omitted from the standard textbooks on special functions in mathematical
Romanovski_polynomials
Family of solutions to related differential equations
}e^{-x\sinh t-\alpha t}\,dt.} The Bessel functions can be expressed in terms of the generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α +
Bessel_function
Mathematical functions
{\mathrm {d} t}{\sqrt {1-t^{4}}}}.} It can also be represented by the hypergeometric function: arcsl x = x 2 F 1 ( 1 2 , 1 4 ; 5 4 ; x 4 ) {\displaystyle \operatorname
Lemniscate_elliptic_functions
Transcendental single-variable function
summation of hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet
Clausen_function
Real root of the polynomial x^5+x+a
differential equations, whose solutions involve hypergeometric functions of several variables. A general formula for differential resolvents of arbitrary
Bring_radical
Mathematical functions
are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and
Falling_and_rising_factorials
Probability distribution
distribution for a real-valued random variable. The general form of its probability density function is f(x)=12πσ2exp(−(x−μ)22σ2). The parameter μ is
Normal_distribution
Special function defined by an integral
where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed
Elliptic_integral
_{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
Special functions used to build correlation functions in 2D CFTs
algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as
Virasoro_conformal_block
Pair of functions in combinatorics
involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and
Wilf–Zeilberger_pair
Irreducible representation of the rotation group SO
) s i m − m ′ , {\displaystyle (-1)^{s}i^{m-m'},} causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of
Wigner_D-matrix
Input to a mathematical function
hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function.
Argument_of_a_function
Mathematical function
( z ) {\displaystyle \sin(z)} . Hypergeometric functions: For p ∈ N {\displaystyle p\in \mathbb {N} } a general formula for a half-integer parameter
Mittag-Leffler_function
Graphical aid for deriving some concepts in combinatorics
objects are not distinguished). This is represented by the generating function 1+1x+1x2+1x3+…=1+x+x2+x3+…=11−x. The series is a geometric series, and
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Family of polynomials
singularium (in Latin). Göttingen: Dieterich. Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood
Gaussian_binomial_coefficient
functions are types of special functions which act as a type of extension from the complete-type of Bessel functions. The incomplete Bessel functions
Incomplete_Bessel_functions
Mathematical functions having established names and notations
a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Many special functions appear
Special_functions
Generalization of the mathematical factorial
mathematics, Bhargava's factorial function, or simply the Bhargava factorial, is a generalization of the factorial function developed by Manjul Bhargava as
Bhargava_factorial
Special mathematical function
polylogarithm of integer order can be expressed as a generalized hypergeometric function: Li n ( z ) = z n + 1 F n ( 1 , 1 , … , 1 ; 2 , 2 , … , 2 ; z
Polylogarithm
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
Computation of an antiderivatives
special functions such as Airy function, error function, Bessel functions, and all hypergeometric functions. A fundamental property of holonomic functions is
Symbolic_integration
System of complete and orthogonal polynomials
generating function for the product of two Legendre polynomials". Norske Videnskabers Selskab Forhandlinger. 29: 82–86. "Sur les séries dont le terme général dépend
Legendre_polynomials
Probability distribution in physics
elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions. The Holtsmark distribution has applications in
Holtsmark_distribution
Canonical solutions of the general Legendre equation
gamma function and 2F1 is the hypergeometric function 2F1(α,β;γ;z)=Γ(γ)Γ(α)Γ(β)∑n=0∞Γ(n+α)Γ(n+β)Γ(n+γ) n!zn, They are called the Legendre functions when
Associated Legendre polynomials
Associated_Legendre_polynomials
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
Russian mathematician (born 1962)
homological algebra. The place of discriminants in the general theory of hypergeometric functions is similar to the place of quasi-classical approximation
Mikhail_Kapranov
Type of probability distribution
distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution. Bayesian programming
Joint probability distribution
Joint_probability_distribution
Polynomial equation of degree 5
quintic equation is one which can be expressed as a quintic function equaling zero. The general form of a quintic equation is a x 5 + b x 4 + c x 3 + d x
Quintic_equation
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
Differential equation that is linear with respect to the unknown function
functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric
Linear_differential_equation
Generating function in integrable systems
the power sum symmetric function. Equation (12) thus shows that the (formal) KP hypergeometric τ {\displaystyle \tau } -function (11) corresponding to the
Tau function (integrable systems)
Tau_function_(integrable_systems)
Soviet mathematician (1913–2009)
combinatorial definition of the Pontryagin class; Coxeter functors; general hypergeometric functions; Gelfand–Tsetlin patterns; Gelfand–Lokutsievski method; the
Israel_Gelfand
{dw}{(1-w^{3})^{2/3}}}} which can also be expressed using the hypergeometric function: sm − 1 ( z ) = z 2 F 1 ( 1 3 , 2 3 ; 4 3 ; z 3 ) {\displaystyle
Dixon_elliptic_functions
Special mathematical function
{\displaystyle |a|<1;\Re (s)<0.} The representation as a generalized hypergeometric function is Φ ( z , s , α ) = 1 α s s + 1 F s ( 1 , α , α , α , ⋯ 1 + α
Lerch_transcendent
Statistical significance test
computational approach relies on a gamma function or log-gamma function, but methods for accurate computation of hypergeometric and binomial probabilities remains
Fisher's_exact_test
Pair of polynomial sequences
yieldsTn(x)=∑j=0⌊n/2⌋(n2j)(x2−1)jxn−2j.This can be written as a 2F1 hypergeometric function: Tn(x)=∑k=0⌊n/2⌋(n2k)(x2−1)kxn−2k=xn∑k=0⌊n/2⌋(n2k)(1−x−2)k=12n
Chebyshev_polynomials
Formal power series
{\sqrt {1+z}}} , the dilogarithm function Li2(z), the generalized hypergeometric functions pFq(...; ...; z) and the functions defined by the power series ∑
Generating_function
Modular function in mathematics
{(A^{2}-3)^{3}}{(A+2)(A-2)}}.} The inverse function of the j-invariant can be expressed in terms of the hypergeometric function 2F1 (see also the article Picard–Fuchs
J-invariant
Identity obeyed by many special functions related to the gamma function
much more common, and follow from characteristic zero relations on the hypergeometric series. The following tabulates the various appearances of the multiplication
Multiplication_theorem
Family of basic hypergeometric orthogonal polynomials in the basic Askey scheme
The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by Q n ( x ; a , b ; q ) = ( a b ;
Al-Salam–Chihara_polynomials
Type of polynomial sequence
factorials" is defined by (x)n=x(x−1)(x−2)⋅⋯⋅(x−n+1). (In the theory of special functions, this same notation denotes upper factorials, but this present usage is
Binomial_type
Data transformation of statistics into rank
approaches offer additional flexibility. One example is the "Rank–rank hypergeometric overlap" approach, which is designed to compare ranking of the genes
Ranking_(statistics)
Operation in calculus
antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending
Integral
(also Hyper) is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial
Petkovšek's_algorithm
Concept in differential equation mathematics
equation has regular singular points at 0, 1 and ∞. A solution is the hypergeometric function. Arscott, F. M. (1995). "Heun's Equation". In Ronveaux, A. (ed
Regular_singular_point
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
Arithmetical function
the Kempner function. Therefore, computing the Kempner function can in general be no easier than factoring composite numbers. Ruler function, a simpler
Kempner_function
Probability distribution
particular instance of the hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution
Student's_t-distribution
Number of subsets of a given size
\alpha } . Binomial transform Delannoy number Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation of
Binomial_coefficient
Chinese-American mathematician
research concerns modular forms, arithmetic hypergeometric functions, as well as number theory in general. She is the Micheal F. and Roberta Nesbit McDonald
Ling_Long_(mathematician)
Mathematical expression
complex-valued hypergeometric functions what are now called Gauss's continued fractions. They can be used to express many elementary functions and some more
Continued_fraction
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
Recursive integer sequence
Stirling's approximation for n ! {\displaystyle n!} , or via generating functions. The only odd Catalan numbers C n {\displaystyle C_{n}} are those
Catalan_number
travel, tourism, insurance
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
Girl/Female
Christian, Gujarati, Indian
Lustrous; Wealthy; Diamond; Rain
Boy/Male
Indian
Lieutenant general
Girl/Female
Shakespearean
Tragedy of King Lear' Daughter to King Lear.
Girl/Female
Christian & English(British/American/Australian)
The Juniper
Female
English
Pet form of French Geneviève, probably GENEVA means "race of women."
Boy/Male
Hindu, Indian
Priceless
Female
Italian
Variant spelling of Italian Ginevra, probably GENEVRA means "race of women."
Boy/Male
Hindu
General nickname
Girl/Female
Assamese, Indian
General
Boy/Male
English French
Surnames derived from place name Deverel.
Girl/Female
Australian, French, Italian
Italian Form of Genevieve; White Wave; Of the Race of Women; Fair and Yielding; Juniper Tree
Girl/Female
Indian, Sanskrit
Brave
Girl/Female
Italian
meaning white wave, of the race of women, fair and yielding.
Girl/Female
Biblical
A wall.
Boy/Male
American, British, English, French
Riverbank; Surnames Derived from Place Name Deverel
Girl/Female
American, Australian, Celtic, Christian, Dutch, French, German, Swiss
Tribe Woman; Of the Race of Women; Juniper Tree; White Wave; Woman; Race of Women; White Race
Girl/Female
French American German
Of the race of women. Juniper.
Boy/Male
Tamil
General nickname
Boy/Male
Muslim
Lieutenant general
Female
Welsh
Medieval Welsh name, probably GENERYS means "white lady."Â
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
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