Search references for GENERALIZED HYPERGEOMETRIC-FUNCTION. Phrases containing GENERALIZED HYPERGEOMETRIC-FUNCTION
See searches and references containing GENERALIZED HYPERGEOMETRIC-FUNCTION!GENERALIZED HYPERGEOMETRIC-FUNCTION
Family of power series in mathematics
{\displaystyle n} is a rational function of n {\displaystyle n} . The series, if convergent, defines a generalized hypergeometric function, which may then be defined
Generalized hypergeometric function
Generalized_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
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
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
Q-analog of hypergeometric series
hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by
Basic_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
Sequence of differential equation solutions
Using the identity 0F1(;α+1;z)=Γ(α+1)z−α/2Iα(2z), (see generalized hypergeometric function), this can also be written as ∑n=0∞n
Laguerre_polynomials
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
Generalization of the hypergeometric function
attempt 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
Meijer_G-function
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
Discrete probability distribution
hypergeometric distributions Negative hypergeometric distribution Multinomial distribution Sampling (statistics) Generalized hypergeometric function Coupon
Hypergeometric_distribution
–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
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
Family of orthogonal polynomials
orthogonal 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,…
Hahn_polynomials
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
Topics referred to by the same term
equation Generalized hypergeometric functions, which generalize the hypergeometric function to specific higher orders General hypergeometric functions, which
Hypergeometric
Mathematical function, denoted exp(x) or e^x
current value of f ( x ) {\displaystyle f(x)} . The exponential function can be generalized to accept complex numbers as arguments. This reveals relations
Exponential_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
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
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
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
μ2+ν2+32;−z24), where 1F2 is a generalized hypergeometric function. Anger function Lommel polynomial Struve function Weber function Lommel (1880). Watson (1922)
Lommel_function
Special function defined by an integral
immediately gives rise to an expression in terms of the generalized hypergeometric function 2 F 2 {\displaystyle {}_{2}\!F_{2}} : Ei ( x ) = x 2
Exponential_integral
Probability distribution
Distribution". Wroughton, Jacqueline. "Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions" (PDF). Hilbe, Joseph M. (2011)
Negative binomial distribution
Negative_binomial_distribution
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
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
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
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
of generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we
Bessel–Clifford_function
that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and
Wilson_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
Sequences. 21 (2) 18.2.7. MR 3779776. Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge University Press. MR 0201688. Steffensen, J. F
Falling_and_rising_factorials
Analytic function that does not satisfy a polynomial equation
zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some
Transcendental_function
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
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
Orthogonal polynomials
by 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 ) = (
Charlier_polynomials
Real root of the polynomial x^5+x+a
Pure Appl. Math. 5: 337–361. Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge University Press. pp. 42–44. ISBN 978-0-521-06483-5
Bring_radical
Special mathematical function
The 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 , …
Polylogarithm
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
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
Polynomial sequence
polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are
Gegenbauer_polynomials
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
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
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
Probability distribution
The exponential generalized beta (EGB) distribution follows directly from the GB and generalizes other common distributions. A generalized beta random variable
Generalized_beta_distribution
Mathematics analytic function
trigonometric and hyperbolic functions. The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which
Hypertranscendental_function
Special mathematical functions defined on the surface of a sphere
More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be
Spherical_harmonics
Well defined hypergeometric series discovered by Giuseppe Lauricella
(corrigendum 1956 in Ganita 7, p. 65) Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X
Lauricella hypergeometric series
Lauricella_hypergeometric_series
Transformation of a mathematical sequence
Paris, 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
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
q\right]} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral
Askey–Wilson_polynomials
this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function: K v ( x , y ) = ∫ 1 ∞ e − x t − y t t v
Incomplete Bessel K function/generalized incomplete gamma function
Incomplete_Bessel_K_function/generalized_incomplete_gamma_function
the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(α) n(x;q) are a family of basic hypergeometric orthogonal polynomials in the
Q-Laguerre_polynomials
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
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_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
Probability distribution
where 1 F 2 {\displaystyle \,_{1}F_{2}} is a generalized hypergeometric function. Hann function Havercosine (hvc) Horst Rinne (2010). "Location-Scale
Raised_cosine_distribution
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 , ⋯
Lerch_transcendent
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
Probability distribution
Bessel function of the first kind. For integer ν {\displaystyle \nu } , the normalizing constant can expressed as a generalized hypergeometric function: Z
Conway–Maxwell–Poisson distribution
Conway–Maxwell–Poisson_distribution
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
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
Mathematical function
Eric W. "Ramanujan Theta Functions". MathWorld. Retrieved 29 April 2018. Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in
Ramanujan_theta_function
1969. He devised the Kampé de Fériet functions, which further generalize the generalized hypergeometric functions. He was an Invited Speaker of the ICM
Joseph_Kampé_de_Fériet
Topics referred to by the same term
free dictionary. PFQ or pFq can refer to: Generalized hypergeometric function, a family of mathematical functions denoted as p F q {\displaystyle _{p}F_{q}}
PFQ
On finite sums of products of three binomial coefficients, and a hypergeometric sum
to an integer) of Dixon's theorem evaluating a well-poised 3F2 generalized hypergeometric series at 1, from (Dixon 1902): 3 F 2 ( a , b , c ; 1 + a − b
Dixon's_identity
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
Mathematical function
a spherical Bessel function. Struve functions (of any order) can be expressed in terms of the generalized hypergeometric function 1F2: H α ( z ) = z α
Struve_function
Mathematical function for the probability a given outcome occurs in an experiment
distributions can be represented with the Dirac delta function as a generalized probability density function f {\displaystyle f} , where f ( x ) = ∑ ω ∈ A p
Probability_distribution
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
Mathematical function
two-parameter Mittag-Leffler function, introduced by Wiman in 1905, is occasionally called the generalized Mittag-Leffler function. It has an additional complex
Mittag-Leffler_function
Special function defined by an integral
in the expression.) Cases of imaginary argument of the generalized integro-exponential function are ∫ 1 ∞ cos ( a x ) ln x x d x = − π 2 24 + γ ( γ
Trigonometric_integral
Probability distribution
plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1F1 and U. E[Xp]=σp⋅(−i2)pU(−p2,12,−μ22σ2),E[|X|p]=σp⋅2p/2Γ(1+p2)π1F1(−p2
Normal_distribution
British mathematician (1922-2008)
January 1922 – 6 June 2008) was a mathematician who worked on hypergeometric functions, and who found many generalizations of the Rogers–Ramanujan identities
Lucy_Joan_Slater
Function related to statistics and probability theory
marginal totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of conditioning is also the basis for Fisher's
Likelihood_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
Indian mathematician
(1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer. ISBN 978-0-387-06482-6. "H-function". "The
Ram_Kishore_Saxena
are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1F1 of one
Humbert_series
Mathematics concept
{1}{2}}}(1/x)} The Bessel polynomial may also be defined as a confluent hypergeometric function y n ( x ) = 2 F 0 ( − n , n + 1 ; ; − x / 2 ) = ( 2 x ) − n U (
Bessel_polynomials
Averages of functions under the Dirichlet distribution
of this type of averaging generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:
Dirichlet_average
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
Special function in mathematics
Φ ( 1 , s , a ) . {\displaystyle \zeta (s,a)=\Phi (1,s,a).\,} Hypergeometric function ζ ( s , a ) = a − s ⋅ s + 1 F s ( 1 , a 1 , a 2 , … a s ; a 1 +
Hurwitz_zeta_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
Mathematical function
\end{aligned}}} With the series expansion of higher rank polygamma function a generalized formula can be given as ∑ n = 0 ∞ u n = ∑ n = 0 ∞ ∑ k = 1 m a k
Digamma_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
^{m+1}(x)P_{n}(\tanh(x))} These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely F n m ( x ) = 3 F
Bateman_polynomials
their properties. The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by p n ( q − x + q x + 1 c d ; a , b
Q-Racah_polynomials
Family of orthogonal polynomials
their properties. The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by P n ( α , β ) ( x ; q ) = ( q n
Continuous q-Jacobi polynomials
Continuous_q-Jacobi_polynomials
Polynomial equation of degree 5
appear at all, and developed his own solution in terms of generalized hypergeometric functions. Similar phenomena occur in degree 7 (septic equations) and
Quintic_equation
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
Hungarian-born British mathematician
Erdélyi was a leading expert on special functions, particularly orthogonal polynomials and hypergeometric functions. He was born Arthur Diamant in Budapest
Arthur_Erdélyi
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
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
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
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
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
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
System of complete and orthogonal polynomials
Associated Legendre polynomials. Legendre functions are solutions of Legendre's differential equation (generalized or not) with non-integer parameters. In
Legendre_polynomials
Difference between logarithm and harmonic series
Kummer Functions ‣ Chapter 11 Confluent Hypergeometric Functions". dlmf.nist.gov. Retrieved 2024-11-01. "DLMF: §9.12 Scorer Functions ‣ Related Functions ‣
Euler's_constant
Algebraic expansion of powers of a binomial
concerned with (x+y)n where the exponent n is a nonnegative integer. The generalized binomial theorem allows for non-integer, negative, or even complex exponents
Binomial_theorem
travel, tourism, insurance
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
GENERALIZED HYPERGEOMETRIC-FUNCTION
travel, tourism, insurance