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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
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
Mathematical function
{s^{b-1}}{(1+s)^{a+b}}}\,ds.} It can also be written in terms of the hypergeometric function: B ( x ; a , b ) = x a a 2 F 1 ( a , 1 − b ; a + 1 ; x ) {\displaystyle
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
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
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
mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Elliptic analog of hypergeometric series
is an elliptic function of n {\displaystyle n} , analogous to generalized hypergeometric series where the ratio is a rational function of n {\displaystyle
Elliptic hypergeometric series
Elliptic_hypergeometric_series
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
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
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
Multivalued function in mathematics
generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r1 = r2, both sides
Lambert_W_function
Generalization of the Meijer G-function and the Fox–Wright function
Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. Mathai, A
Fox_H-function
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
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
Polynomial sequence
Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n
Jacobi_polynomials
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
In physics, solution to Schrödinger equation
potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb wave equation for
Coulomb_wave_function
Generalisation of the generalised hypergeometric function pFq(z)
function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric
Fox–Wright_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
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
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_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
Laguerre_polynomials
Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series
List of eponyms of special functions
List_of_eponyms_of_special_functions
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
function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function
List of mathematical functions
List_of_mathematical_functions
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
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
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
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
Set of four hypergeometric series
of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable
Appell_series
In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman
Bateman_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
of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists
List of hypergeometric identities
List_of_hypergeometric_identities
In mathematics, a solution to a modified form of the confluent hypergeometric equation
mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by
Whittaker_function
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 much
Wilf–Zeilberger_pair
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
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
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
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
Special function defined by an integral
{i^{k}}{(m+nk+1)}}{\frac {x^{m+nk+1}}{k!}}} is a confluent hypergeometric function and also an incomplete gamma function ∫ x m e i x n d x = x m + 1 m + 1 1 F 1 ( m
Fresnel_integral
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
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
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
Type of functions, in mathematical analysis
the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions. Examples of
Holonomic_function
Summation method for hypergeometric terms
where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given
Gosper's_algorithm
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
Family of orthogonal polynomials
polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by Q n ( x ; α , β , N ) = 3 F 2 ( − n , − x , n + α + β + 1 ;
Hahn_polynomials
Concept in mathematics
) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear
Parabolic_cylinder_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
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
generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have
Bessel–Clifford_function
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
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
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 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
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
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
System of complete and orthogonal polynomials
polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. In this approach,
Legendre_polynomials
Risk measure estimating the average loss in the worst tail of the distribution
beta function is defined only for positive arguments, for a more generic case the expected shortfall can be expressed with the hypergeometric function: ES
Expected_shortfall
Probability distribution
characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first
Beta_distribution
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
Special function defined by an integral
the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic
Trigonometric_integral
Pair of polynomial sequences
This can be written as a 2 F 1 {\displaystyle {}_{2}F_{1}} hypergeometric function: T n ( x ) = ∑ k = 0 ⌊ n / 2 ⌋ ( n 2 k ) ( x 2 − 1 ) k x n − 2
Chebyshev_polynomials
–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 − n 2
Mott_polynomials
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
Topics referred to by the same term
G-function, related to the Gamma function Meijer G-function, a generalization of the hypergeometric function Siegel G-function, a class of functions in
G-function
Special function defined by an integral
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number
Logarithmic_integral_function
Measure of linear correlation
is the gamma function and 2 F 1 ( a , b ; c ; z ) {\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)} is the Gaussian hypergeometric function. In the special
Pearson correlation coefficient
Pearson_correlation_coefficient
Real root of the polynomial x^5+x+a
series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation x 5 + x + a = 0 {\displaystyle
Bring_radical
German polymath and scholar (1777–1855)
the general hypergeometric function F ( α , β , γ , x ) {\displaystyle F(\alpha ,\beta ,\gamma ,x)} , and shows that many of the functions known at the
Carl_Friedrich_Gauss
Algebraic expansion of powers of a binomial
{n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ (
Binomial_theorem
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 ; q
Jackson_q-Bessel_function
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ℓ = 10 {\displaystyle \ell =10} . Some of these
Table_of_spherical_harmonics
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
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
Japanese mathematician
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
Measure of internal forces in an atomic nucleus
}{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}} is the hypergeometric function. It is also possible to analytically solve the eigenvalue problem
Woods–Saxon_potential
Triangular array of the binomial coefficients
convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent
Pascal's_triangle
Polynomial sequence
{n-m}{2}}-k}}\rho ^{n-2k}} . A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special
Zernike_polynomials
Monochrome light beam whose amplitude envelope is a Gaussian function
real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can
Gaussian_beam
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
Transformation of a mathematical sequence
R. B. (2010). "Euler-type transformations for the generalized hypergeometric function". Z. Angew. Math. Phys. 62 (1): 31–45. doi:10.1007/s00033-010-0085-0
Binomial_transform
their properties. The polynomials are given in terms of the basic hypergeometric function by C n ( q − x ; a ; q ) = 2 ϕ 1 ( q − n , q − x ; 0 ; q , − q
Q-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
generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero,
Egorychev_method
Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by p n ( t 2 ) = ( a + b ) n ( a + c
Wilson_polynomials
Probability distribution
Distribution". Wroughton, Jacqueline. "Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions" (PDF). Hilbe, Joseph M. (2011)
Negative binomial distribution
Negative_binomial_distribution
Integral transform
wave equation called John's equation. The Gaussian or ordinary hypergeometric function can be written as an X-ray transform (Gelfand, Gindikin & Graev
X-ray_transform
Number theory expression
{n}{p^{i}}}\right\rfloor ,} where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } is the floor function. While the sum on the right side is an infinite sum, for any particular
Legendre's_formula
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
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
Graphical aid for deriving some concepts in combinatorics
objects are not distinguished). This is represented by the generating function 1 + 1 x + 1 x 2 + 1 x 3 + … = 1 + x + x 2 + x 3 + … = 1 1 − x . {\displaystyle
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Solutions of the Laplace equation in spherical polar coordinates
Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } . There are two
Solid_harmonics
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
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
In mathematics, the Toronto function is a modification of the confluent hypergeometric function defined by A. H. Heatley as T ( m , n , r ) = r 2 n −
Toronto_function
Generalization of the binomial theorem to other polynomials
Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, log ( k n n , n , ⋯ , n ) = k n log ( k ) +
Multinomial_theorem
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HYPERGEOMETRIC FUNCTION
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HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
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HYPERGEOMETRIC FUNCTION
HYPERGEOMETRIC FUNCTION
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