Search references for GENERAL HYPERGEOMETRIC-FUNCTION. Phrases containing GENERAL HYPERGEOMETRIC-FUNCTION
See searches and references containing GENERAL HYPERGEOMETRIC-FUNCTION!GENERAL HYPERGEOMETRIC-FUNCTION
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
a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series
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
of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists
List of hypergeometric identities
List_of_hypergeometric_identities
Special function in mathematics
In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de
Kampé_de_Fériet_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 ,
Carl_Friedrich_Gauss
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
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
Mathematical series
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 function of n. The
Bilateral hypergeometric series
Bilateral_hypergeometric_series
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
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
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
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
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
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
Probability distribution
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 [ X
Normal_distribution
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
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
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
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
Solutions of Legendre's differential equation
expressed in terms of the hypergeometric function, 2 F 1 {\displaystyle _{2}F_{1}} . With Γ {\displaystyle \Gamma } being the gamma function, the first solution
Legendre_function
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_polynomials
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
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
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
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
Soviet mathematician (1913–2009)
combinatorial definition of the Pontryagin class; Coxeter functors; general hypergeometric functions; Gelfand–Tsetlin patterns; Gelfand–Lokutsievski method; the
Israel_Gelfand
Polynomial function of degree 5
at all, and developed his own solution in terms of generalized hypergeometric functions. Similar phenomena occur in degree 7 (septic equations) and 11
Quintic_function
Linear recurrence equation
y ( n ) {\textstyle y(n)} is called hypergeometric if the ratio of two consecutive terms is a rational function in n {\displaystyle n} , i.e. y ( n +
P-recursive_equation
Mathematics concept
version of the hypergeometric differential equation Curiously, they have been omitted from the standard textbooks on special functions in mathematical
Romanovski_polynomials
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
Infinite sum
convergence tests. As a function of p {\displaystyle p} , the sum of this series is Riemann's zeta function. Hypergeometric series: p F q [ a 1 , a
Series_(mathematics)
_{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
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
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
Operation in mathematical calculus
antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending
Integral
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
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
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
Mathematical theorem on convolved binomial coefficients
{\displaystyle \;_{2}F_{1}} is the hypergeometric function and Γ ( n + 1 ) = n ! {\displaystyle \Gamma (n+1)=n!} is the gamma function. One regains the Chu–Vandermonde
Vandermonde's_identity
Class of differential equations expressible in differential algebra
operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely
Algebraic differential equation
Algebraic_differential_equation
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
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
Canonical solutions of the general Legendre equation
{\displaystyle \Gamma } 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 (
Associated Legendre polynomials
Associated_Legendre_polynomials
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
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
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
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
Nonlinear differential operator used to study conformal mappings
the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller
Schwarzian_derivative
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
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
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)
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
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
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
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
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
Pattern defining an infinite sequence of numbers
exponential function of an integral. Many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series. Special
Recurrence_relation
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)
{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
Mathematical concept
fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to
Gauss's_continued_fraction
transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by where
Zonal_spherical_function
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
Function related to statistics and probability theory
probability distributions (a more general definition is discussed below). Given a probability density or mass function x ↦ f ( x ∣ θ ) , {\displaystyle
Likelihood_function
Special function occurring in problems possessing elliptic symmetry
Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic
Mathieu_function
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 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
Mathematical function
poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto. Aomoto proved a slightly more general integral
Selberg_integral
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)
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
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 analysis
Bernhard_Riemann
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
Probability distribution
1F1 is the confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated as
Wigner semicircle distribution
Wigner_semicircle_distribution
Russian-American mathematician
other contributions include: Contributions to the theory of general hypergeometric functions Contributions to the theory of Lie–Massey operators Instigated
Vladimir_Retakh
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
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
Boy/Male
Hindu, Indian
Priceless
Girl/Female
French American German
Of the race of women. Juniper.
Boy/Male
American, British, English, French
Riverbank; Surnames Derived from Place Name Deverel
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
General nickname
Girl/Female
Christian, Gujarati, Indian
Lustrous; Wealthy; Diamond; Rain
Boy/Male
Indian
Lieutenant general
Boy/Male
Tamil
General nickname
Girl/Female
Australian, French, Italian
Italian Form of Genevieve; White Wave; Of the Race of Women; Fair and Yielding; Juniper Tree
Girl/Female
Assamese, Indian
General
Girl/Female
Shakespearean
Tragedy of King Lear' Daughter to King Lear.
Female
Italian
Variant spelling of Italian Ginevra, probably GENEVRA means "race of women."
Female
Welsh
Medieval Welsh name, probably GENERYS means "white lady."Â
Girl/Female
Indian, Sanskrit
Brave
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
Italian
meaning white wave, of the race of women, fair and yielding.
Boy/Male
Muslim
Lieutenant general
Girl/Female
Biblical
A wall.
Boy/Male
English French
Surnames derived from place name Deverel.
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
Boy/Male
Tamil
Gunaratna | கà¯à®¨à®¾à®°à®¤à¯à®¨à®¾
Jewel of virtue
Boy/Male
Hindu
Lotus
Boy/Male
Hindu, Indian
Judicious
Boy/Male
Anglo, British, English
Name of an Earl
Girl/Female
Hindu
Boy/Male
Hindu
Hemaansh = a part of gold
Boy/Male
Hindu
Sharpness, Brightness
Girl/Female
Indian
Pure, Beautiful
Boy/Male
English French
Open.
Boy/Male
Muslim
God is my father
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
GENERAL HYPERGEOMETRIC-FUNCTION
a.
Arising from sexual intercourse; as, a venereal disease; venereal virus or poison.
a.
Common to many, or the greatest number; widely spread; prevalent; extensive, though not universal; as, a general opinion; a general custom.
a.
Adapted to the cure of venereal diseases; as, venereal medicines.
adv.
In general; commonly; extensively, though not universally; most frequently.
pl.
of Postmaster-general
a.
Not restrained or limited to a precise import; not specific; vague; indefinite; lax in signification; as, a loose and general expression.
n. pl.
Generalities; general terms.
n.
Gum senegal. See under Gum.
a.
The roll of the drum which calls the troops together; as, to beat the general.
v. i.
Anything which is neither animal nor vegetable, as in the most general classification of things into three kingdoms (animal, vegetable, and mineral).
adv.
In a general way, or in general relation; in the main; upon the whole; comprehensively.
a.
acting as a generant.
a.
Alt. of Generical
a.
Comprehending many species or individuals; not special or particular; including all particulars; as, a general inference or conclusion.
a.
Usual; common, on most occasions; as, his general habit or method.
n.
The venereal disease; syphilis.
a.
Having a relation to all; common to the whole; as, Adam, our general sire.
a.
Relating to a genus or kind; pertaining to a whole class or order; as, a general law of animal or vegetable economy.