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Hypergeometric may refer to several distinct concepts within mathematics: The hypergeometric function, a solution to the Gaussian hypergeometric differential...

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k {\displaystyle...

the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many...

In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function...

mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in...

a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential...

list of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function...

In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by Mary Celine Fasenmyer in 1947. They include Legendre...

1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several q-analogs such as the Jackson–Bessel functions...

In mathematics, an elliptic hypergeometric series is a series ∑ c n {\displaystyle {\textstyle \sum c_{n}}} such that the ratio c n c n − 1 {\displaystyle...

William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation...

theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities...

Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet. The Kampé de Fériet function...

In statistics, the hypergeometric distribution is the discrete probability distribution generated by picking colored balls at random from an urn without...

In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without...

following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand...

Wallenius' noncentral hypergeometric distribution (named after Kenneth Ted Wallenius) is a generalization of the hypergeometric distribution where items...

mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These...

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...

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are...

and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions. Let F {\textstyle \mathbb {F} } be a field...

sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent...

mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced...

the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Hahn...

American mathematician whose research concerns modular forms, arithmetic hypergeometric functions, as well as number theory in general. She is the Micheal F...

–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 −...

generalized Stieltjes–Wigert polynomials P(α) n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Moak...

scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials...

gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified...

mathematician and Catholic religious sister. She is most noted for her work on hypergeometric functions and linear algebra. Fasenmyer grew up in Pennsylvania's oil...

four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series...

function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials & approximations Factorial...

In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio an/an+1 of two terms is a rational...

theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 −...

In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by...

Kummer's function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions...

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...

higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation...

introduced Bailey's lemma and Bailey pairs into the theory of basic hypergeometric series. Bailey chains and Bailey transforms are named after him. Slater...

Aomoto is a Japanese mathematician who introduced the Aomoto-Gel'fand hypergeometric function and the Aomoto integral. He was a professor at Nagoya University...

mathematician and writer. He specialized in fields of mathematics such as hypergeometric series and orthogonal polynomials. He also had interests encompassing...

Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized...

{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...

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...

class of Appell polynomials can be obtained in terms of the generalized hypergeometric function. Let Δ ( k , − n ) {\displaystyle \Delta (k,-n)} denote the...

polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by S n ( x 2 ; a ...

Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that 2 F 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c −...

equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur...

{1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L n ( α ) ( x ) := ( n + α...

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...

sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem...

Russian mathematician and number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and...

{z^{n}}{n!}}} and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and...

Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers. Define μ {\displaystyle...

Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two (i.e. having two independent variables), enumerated...

Exponential F Gamma Geometric Gumbel Hypergeometric Laplace Logistic Log-normal Logarithmic Multinomial Multivariate hypergeometric Multivariate normal Negative...

which organizes orthogonal polynomials of ( q {\displaystyle q} -)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials...

numbers that are picked on each ticket. Keno probabilities come from a hypergeometric distribution. For Keno, one calculates the probability of hitting exactly...

{\displaystyle y} , in an ordered pair ( x , y ) {\displaystyle (x,y)} . The hypergeometric function is an example of a four-argument function. The number of arguments...

} is the gamma function and 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function 2 F 1 ( α , β ; γ ; z ) = Γ ( γ ) Γ ( α ) Γ ( β ) ∑ n = 0 ∞...

Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Given an infinite series...

by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table. This setting...

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function...

another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately...

{\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0} , (f periodic) Physics Hypergeometric 2 z ( 1 − z ) d 2 w d z 2 + [ c − ( a + b + 1 ) z ] d w d z − a b w...

the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer...

polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 (Chebyshev...

(also Hyper) is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial...

phylogeny or evolutionary relationships. It results in a measure of hypergeometric "distance" between taxa. Phenetic methods have become relatively rare...

}&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson...

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...

Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n)...

there) working on hypergeometric functions, who introduced the Hahn–Exton q-Bessel function. Exton, Harold (1976), Multiple hypergeometric functions and applications...

1888. F. J. Whipple coined the phrase "Saalschützian" for generalized hypergeometric 3 F 2 {\displaystyle _{3}F_{2}} series where one of the numerator parameters...

Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series. He is best known for his lecture notes on the subject which...

the plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1 F 1 {\textstyle {}_{1}F_{1}} and U . {\textstyle U.} E ⁡...

{163}}}{2}}\right)=-640320^{3}} , and on the following rapidly convergent generalized hypergeometric series: 1 π = 10005 4270934400 ∑ k = 0 ∞ ( − 1 ) k ( 6 k ) ! ( 545140134...

functions (Handbuch der Kugelfunctionen). He also investigated basic hypergeometric series. He introduced the Mehler–Heine formula. Heinrich Eduard Heine...

series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. Given a series S ( a , b ) = ∑ n = a b p...

polynomials and basic hypergeometric series, who introduced the Askey–Gasper inequality. Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia...

Randomness Model is based on the Bernoulli model and its limiting forms, the hypergeometric distribution, Bose–Einstein statistics and its limiting forms, the compound...

(n)_{r}} are understood to be zero if r > n. If a random variable X has a hypergeometric distribution with population size N, number of success states K ∈ {0...

stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as V = V 0 1 + W ( e − x σ ) . {\displaystyle...

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...

Goursat also published texts on partial differential equations and hypergeometric series. Edouard Goursat was born in Lanzac, Lot. He was a graduate of...

{\displaystyle n\to \infty } . With 2 F 1 {\displaystyle {}_{2}F_{1}} being the hypergeometric function: ∑ n = 0 ∞ r 2 ( n ) q n = 2 F 1 ( 1 2 , 1 2 , 1 , z ) {\displaystyle...

Rutgers University. Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. He gave the first proof of the alternating...

solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving...

mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike...

mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz...

hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution...

fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known...

number of draws before the first successful (correctly colored) draw. hypergeometric distribution: the balls are not returned to the urn once extracted....

particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...

the identification of community structures. The Casiraghi-Nanumyan Hypergeometric Configuration Model extends canonical configuration models by accounting...

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...

\right)\ } where   F H y p   {\displaystyle \ F_{Hyp}\ } is the Gaussian hypergeometric function. This density is both a Bayesian posterior density and an exact...

mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

hypergeometric functions Gauss's criterion – described on Encyclopedia of Mathematics Gauss's hypergeometric theorem, an identity on hypergeometric series...

1007/s10955-005-7583-z. Rains, Eric M. (2010). "Transformations of elliptic hypergeometric integrals". Annals of Mathematics. 171 (1): 169–243. doi:10.4007/annals...

{\displaystyle \alpha } ⁠. Binomial transform Delannoy number Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation...

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number...

mathematician who studies number theory, noncommutative geometry, and hypergeometric functions. She is a professor emerita of mathematics at Texas A&M University...

increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun,...

Slater (5 January 1922 – 6 June 2008) was a mathematician who worked on hypergeometric functions, and who found many generalizations of the Rogers–Ramanujan...

approaches offer additional flexibility. One example is the "Rank–rank hypergeometric overlap" approach, which is designed to compare ranking of the genes...

known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. q-analogs are most...

statistics, zonal polynomials, distance correlation, total positivity, and hypergeometric functions of matrix argument. He is a distinguished professor emeritus...

\mathbf {s} } variables, known as a τ {\displaystyle \tau } -function of hypergeometric type. In particular, choosing r j = r j β := e j β {\displaystyle r_{j}=r_{j}^{\beta...

Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. (Srivastava & Manocha...

the University of Zaragoza and since then published extensively on hypergeometric identities, WZ-pairs, and related topics in analytic number theory....

expert on special functions, particularly orthogonal polynomials and hypergeometric functions. He was born Arthur Diamant in Budapest, Hungary to Ignác...

Ankara, Turkey) is a Dutch mathematician, who works on number theory and hypergeometric functions. In 1979 Beukers received his PhD at Leiden University under...

Kampé de Fériet functions, which further generalize the generalized hypergeometric functions. He was an Invited Speaker of the ICM in 1928 at Bologna,...

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...

superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Computer Aided Geometric Design 29(7): 510–518, 2012 [9]....

} The Bessel functions can be expressed in terms of the generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α + 1 ) 0 F 1 ( − ; α + 1 ; − x...

mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

{n-2k}{{\tfrac {n-m}{2}}-k}}\rho ^{n-2k}} . A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they...

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A...

recover or succumb to the illness. The third design is given by the hypergeometric distribution; where both the total numbers in each column and row are...

_{2}\left(\alpha ;\gamma ,\gamma ';x,y\right)} is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of...

Kapranov investigated generalized Euler integrals, A {\displaystyle A} -hypergeometric functions, A {\displaystyle A} -discriminants, and hyperdeterminants...

of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated...

the theory of binary and ternary quadratic forms, and the theory of hypergeometric series. When Gauss was only 19 years old, he proved the construction...

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and...

(3). doi:10.37236/2481. Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed...

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...

equations of even order. Weyl applied his theory to Carl Friedrich Gauss's hypergeometric differential equation, thus obtaining a far-reaching generalisation...

Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions...

In mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function...

O{\mathord {\left(M(n)n^{1/2}(\log n)^{2}\right)}}} Fixed rational number Hypergeometric series O ( M ( n ) ( log ⁡ n ) 2 ) {\displaystyle O{\mathord {\left(M(n)(\log...

ISBN 978-0-8176-4566-3. Schechtman, V. V.; Varchenko, A. N. (1990). "Hypergeometric solutions of Knizhnik-Zamolodchikov equations". Lett. Math. Phys. 20...

\,} This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called...

{d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0} is usually solved by the confluent hypergeometric functions M ( a , b , z ) {\displaystyle M(a,b,z)} and ⁠ U ( a , b ...

functions; rather, the probability density function is expressed in terms of hypergeometric functions. The Holtsmark distribution has applications in plasma physics...

{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...

multivariate normal, other elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial, or Dirichlet distribution, but not...

mulea calculates a p-value ( p j ) {\textstyle (p_{j})} based on the hypergeometric test. To assess the unbiased statistical significance of each ontology...

projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions...

and zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for...

particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb...

Springer. ISBN 978-1-4612-7221-2. p. 326 This formula can be proved by hypergeometric inversion: Let a ⁡ ( q ) = ∑ m , n ∈ Z q m 2 + m n + n 2 {\displaystyle...

{3}{2}};3;2iRt\right)={\frac {2J_{1}(Rt)}{Rt}},} where 1F1 is the confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the...

In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

linear recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the...

; Garvan, Frank (eds.). Analytic Number Theory, Modular Forms and q-Hypergeometric Series. Springer Proceedings in Mathematics & Statistics. Vol. 221....

hypergeometric series, dating back to 1837. which cites to Kummer, Ernst Eduard (1836). "Uber die Hypergeometrische Reihe" [About the hypergeometric series]...

here by 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 −...

doi:10.37236/1237. Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", Electronic Journal of Combinatorics...

)^{-1/k}\right)\right]} , where 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function. Alternatively, ES α ⁡ ( X ) = − γ − β α c k c + 1 ( ( 1 −...

In mathematics, Lentz's algorithm is an algorithm to evaluate continued fractions, and was originally devised to compute tables of spherical Bessel functions...

superset of the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions...

MathWorld. Retrieved 29 April 2018. Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol...

ultrahyperbolic wave equation called John's equation. The Gaussian or ordinary hypergeometric function can be written as an X-ray transform.(Gelfand, Gindikin & Graev...

{dw}{(w-1)^{1-a}(w+1)^{1-b}}},} which can be expressed in terms of hypergeometric functions or incomplete beta functions. The upper half-plane is mapped...

an advanced researcher in fields, such as mock theta functions and hypergeometric series, related closely to works of Ramanujan. In 1970, anticipating...

linearly with the position d. D. J. Broadhurst, "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)", (1998) arXiv...

much more common, and follow from characteristic zero relations on the hypergeometric series. The following tabulates the various appearances of the multiplication...

Ono showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight ⁠3/2⁠ harmonic weak...

developed by Carl Gustav Jacob Jacobi. The equation had the form of a hypergeometric equation to which Jacobi had found a solution. Kronig and Rabi wrote...

converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions. Since...

use these definitions to demonstrate a number of properties of the hypergeometric function. Although Díaz and Pariguan restrict these symbols to k > 0...

F5 algorithm) Gosper's algorithm: find sums of hypergeometric terms that are themselves hypergeometric terms Knuth–Bendix completion algorithm: for rewriting...

{\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...

functions such as Airy function, error function, Bessel functions, and all hypergeometric functions. A fundamental property of holonomic functions is that the...

A spectral type F represents force in mechanics equations pFq is a hypergeometric series the probability distribution function in statistics a Fibonacci...

Ernst Eduard (1836). "Uber die Hypergeometrische Reihe" [About the hypergeometric series]. Journal für die Reine und Angewandte Mathematik (in German)...

{\displaystyle {\sqrt {1+z}}} , the dilogarithm function Li2(z), the generalized hypergeometric functions pFq(...; ...; z) and the functions defined by the power series...

(1991) Fallen angel (2001) Hypergeometrical composition of A1 (2012) Hypergeometrical composition No. 4 (2012) Hypergeometrical composition No. 5 (2012)...

_{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...

Symbolic integration via e.g. Risch algorithm or Risch–Norman algorithm Hypergeometric summation via e.g. Gosper's algorithm Limit computation via e.g. Gruntz's...

representations of real numbers and Gosper's algorithm for finding closed form hypergeometric identities. In 1985, Gosper briefly held the world record for computing...

Distribution". Wroughton, Jacqueline. "Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions" (PDF). Hilbe, Joseph M. (2011)...

{\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!} (see Confluent hypergeometric function) Multivariate normal N ( μ , Σ ) {\displaystyle N(\mathbf {\mu...

branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions 2F1(a, b; c; z) and the incomplete beta function B(z; a, b):...

distribution q-Weibull distribution Tsallis q-Gaussian Tsallis entropy Basic hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson...

Iwahori–Hecke algebras, with hypergeometric functions associated with Lie algebra root systems (Heckman-Opdam hypergeometric functions), and with Dunkl...

in 2009 at the University of Texas at Austin, with the dissertation Hypergeometric Functions in Arithmetic Geometry supervised by Fernando Rodríguez-Villegas...

In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions, and the analytic...

Function The Zeta Function of Riemann The Hypergeometric Function Legendre Functions The Confluent Hypergeometric Function Bessel Functions The Equations...

1016/S0169-7439(03)00029-7. de Boor, p. 115. Carlson, B.C. (1991). "B-splines, hypergeometric functions, and Dirichlet averages". Journal of Approximation Theory...

family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. The polynomials are given in terms of basic hypergeometric functions by...

(PDF) on 28 November 2009. Retrieved 9 March 2022. Exton, H. (1983). q-Hypergeometric Functions and Applications. New York: Halstead Press. ISBN 978-047027453-8...

{1}{2}};{\frac {1+r\rho }{2}}\right)} where F {\displaystyle F} is the Gaussian hypergeometric function and ν = N − 1 > 1 {\displaystyle \nu =N-1>1} . While the Fisher...

convergence of an alternating series and is also frequently applied to the hypergeometric series Euler rotation equations, a set of first-order ODEs concerning...

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)...

transformation in the Cremona group Kummer's quadratic transformation of the hypergeometric function This disambiguation page lists mathematics articles associated...

polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by p n ( x ; a , b...

appear at all, and developed his own solution in terms of generalized hypergeometric functions. Similar phenomena occur in degree 7 (septic equations) and...

Shigeru (2011). "10 trillion digits of pi: A case study of summing hypergeometric series to high precision on multicore systems" (PDF). Alexander Jih-Hing...

sequence, public-key crypto functions, elliptic curves, fractal dimension, hypergeometric functions, Fibonacci polytopes, geometry, graph theory, music, and art...

John (1937). "Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions" (PDF). Annals of Mathematical Statistics. 8...

Liouvillian include: the Bessel functions (except special cases); the hypergeometric functions (except special cases). Examples of functions which are not...

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...

identities: Combinatorial Fibonacci identities and Other Fibonacci identities Hypergeometric function identities List of integrals of logarithmic functions List...

distribution. For a simple random sample without replacement, one obtains a hypergeometric distribution. Several efficient algorithms for simple random sampling...

Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k ν (...

The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling...

this potential can be solved analytically, by transforming it into a hypergeometric differential equation. The radial part of the wavefunction solution...

In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic...

In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through...

characteristic function of the beta distribution is Kummer's confluent hypergeometric function (of the first kind): φ X ( α ; β ; t ) = E ⁡ [ e i t X ] =...

milk first follows the hypergeometric distribution   Hypergeometric ( 8 , 4 , 4 )   . {\displaystyle \ {\mbox{Hypergeometric}}(8,4,4)~.} Boschloo's test...

2 ] {\textstyle C_{n}=\pi ^{n/2}/\Gamma [1+{\frac {n}{2}}]} and the hypergeometric function 2 F 1 {\displaystyle {}_{2}F_{1}} or the regularized incomplete...

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...

2,.... The Euler transform is also frequently applied to the Euler hypergeometric integral 2 F 1 {\displaystyle \,_{2}F_{1}} . Here, the Euler transform...

polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice x...

orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q...

with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland...

the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered...

distribution, the hyperexponential distribution is not analogous to the hypergeometric distribution. The hyperexponential distribution is an example of a mixture...

systems and hypergeometric functions. II", Compositio Mathematica, 64 (3): 353–373, MR 0918417 Opdam, E. M. (1988), "Root systems and hypergeometric functions...

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...

combinatorial definition of the Pontryagin class; Coxeter functors; general hypergeometric functions; Gelfand–Tsetlin patterns; Gelfand–Lokutsievski method; the...

}{(1-\nu \sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}} Confluent hypergeometric functions double conf_hyperg(double a, double c, double x); F ( a ,...

where 1 F 2 {\displaystyle \,_{1}F_{2}} is a generalized hypergeometric function. Hann function Havercosine (hvc) Horst Rinne (2010). "Location-Scale...

logarithm, and the trigonometric and hyperbolic functions. The generalized hypergeometric functions, including special cases such as Bessel functions (except...

after him: one for the sum of a 7F6 hypergeometric series, and another for the sum of a bilateral hypergeometric series. Dougall was born in June 1867...

University of the Witwatersrand. Her 2001 dissertation, Zeros of general hypergeometric polynomials, was supervised by Kathy Driver. She was a member of the...

) {\displaystyle {}_{1}F_{1}(\cdot ;\cdot ,\cdot )} is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of...

singularity theory, braid theory, reflection groups, invariant theory, and hypergeometric integrals. He was, with Louis Solomon and Hiroaki Terao, a pioneer of...

geometry Quantum differential calculus Time scale calculus q-analog Basic hypergeometric series Quantum dilogarithm Abreu, Luis Daniel (2006). "Functions q-Orthogonal...

serierum singularium (in Latin). Göttingen: Dieterich. Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis...

)}}\end{aligned}}} Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary...

A.; Solomon, A. I. (2001). "Extended Bell and Stirling Numbers From Hypergeometric Exponentiation" (PDF). Journal of Integer Sequences. 4: 01.1.4. arXiv:math/0106123...

out that it occurred in the unpublished note by Riemann (1863) about hypergeometric functions. The contour of steepest descent has a minimax property, see...

polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam...

gathered. Molenaar, W. (1970). Approximations to the Poisson, Binomial and Hypergeometric Distribution Functions. Mathematisch Centrum. ISBN 978-90-6196-053-9...

value of the parameter". Fisher's noncentral hypergeometric distribution, a generalization of the hypergeometric distribution, where sampling probabilities...

triple product formula, as well as many other results on q-series and hypergeometric series. The solution of the Jacobi inversion problem for the hyperelliptic...

solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation...

the expectation of the local statistics are available based on the hypergeometric distribution but due to the multiple comparisons problem a permutation...

introduced the MacRobert E function, a generalisation of the generalised hypergeometric series. He was born on 4 April 1884 in the manse at Dreghorn, Ayrshire...

Died (1983-05-06)6 May 1983 Moscow, Russia Known for Special functions Hypergeometric functions Awards N T Veatch award for Distinguished Research and Creative...

1797a. Pfaff 1797b. Jacques Dutka (1984). "The Early History of the Hypergeometric Function". Archive for History of Exact Sciences. 31 (1): 15–34. JSTOR 41133728...

finite support Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot...

}{\frac {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...

degree k, and it is a hypergeometric series with rational coefficients (the precise coefficients can be derived using the hypergeometric series for Jacobi...

Calabi–Yau complete intersections using the generalized A {\displaystyle A} -hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei...

into a block matrix). For example, a classical question was for the hypergeometric equation: when does it have a pair of algebraic solutions, in terms...

introduced by Meixner Q-Meixner polynomials, are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Q-Meixner–Pollaczek...

introduced the Pochhammer symbol, now generally used for expressing hypergeometric functions in a compact notation. Pochhammer was born in Stendal, but...

a set of four hypergeometric series F1, F2, F3, F4 of two variables, now called Appell series, that generalize Gauss's hypergeometric series. He established...

finite support Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot...

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions...

harmonics, factorials and gamma functions, zeta functions, polynomials, hypergeometric, special functions, etc. Substitution Arbitrary precision integers,...

other contributions include: Contributions to the theory of general hypergeometric functions Contributions to the theory of Lie–Massey operators Instigated...

IdeaCentre Q series, nettop computers Pentax Q series, cameras Q-series Hypergeometric q-series Q (disambiguation) This disambiguation page lists articles...

first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions. Euler's constant has been among the mathematical constants...

Galois and C. Jordan. In the section on automorphic forms, he treats hypergeometric functions, conformal mappings, the icosahedron and elliptic functions...

where ⁠ U ( a , b , z ) {\displaystyle U(a,b,z)} ⁠ is the confluent hypergeometric function of the second kind. In instances where the F-distribution is...

In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of J ν ( x e 3 π i 4 ) , {\displaystyle...

ISBN 978-0-88385-547-8. Rao, K. S.; Berghe, G. V. (2003). "Gauss, Ramanujan and Hypergeometric Series Revisited". Historia Scientiarum. 13 (2): 123–133. Schmadel,...

z ) {\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)} is the Gaussian hypergeometric function. In the special case when ρ = 0 {\displaystyle \rho =0} (zero...

coefficient Gould's sequence Hyperfactorial Hypergeometric distribution Hypergeometric function identities Hypergeometric series Incomplete beta function Incomplete...

Gauss's principle of least constraint Gauss's digamma theorem Gauss's hypergeometric theorem Gaussian function  See also: List of things named after Carl...

inverse function of the j-invariant can be expressed in terms of the hypergeometric function 2F1 (see also the article Picard–Fuchs equation). Explicitly...

below. The Romanovski polynomials solve the following version of the hypergeometric differential equation Curiously, they have been omitted from the standard...

MR 1313873 van Diejen, Jan F. (1999), "Properties of some families of hypergeometric orthogonal polynomials in several variables", Trans. Amer. Math. Soc...

Q_{j})=B(P_{i},Q_{i})\delta _{ij}} . Appell sequence Askey scheme of hypergeometric orthogonal polynomials Favard's theorem Polynomial sequences of binomial...

; z ) {\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...

2 {\displaystyle i=1,2} , O i , j {\displaystyle O_{i,j}} follows a hypergeometric distribution with parameters N j {\displaystyle N_{j}} , N i , j {\displaystyle...

Janisch, Hyunju Kwon 2024-02-13 148 9780691257532 220 Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups Nicholas M. Katz, Pham Huu Tiep 2025-06-24...

In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A....

available code for manipulating polynomials over GF(2), and for calculating hypergeometric constants to billions of decimal places. He is associated with the CARAMEL...

to be basic. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated...

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...

functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm...

in Univariate Discrete Distribution Theory Based on the Generalized Hypergeometric Function and Associated Differential Equations. The family moved again...

integral M24: Filtering sigmoid functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions...

most four regular singular points, such as the Lamé equation or the hypergeometric differential equation, can be transformed into this equation by a change...

Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer...

been used to fit data including modeling patients spreading COVID-19. Hypergeometric distribution Coupon collector's problem Compound Poisson distribution...

doi:10.1093/jee/84.1.140. Shiyomi M, Egawa T, Yamamoto Y (1998) Negative hypergeometric series and Taylor's power law in occurrence of plant populations in...

groups and root systems Enumerative combinatorics Symmetric functions Hypergeometric series and q-series Computational problems and algorithms in algebra...

Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function: K v ( x , y...

antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on)...

Bessel functions) and ν = 1 / 3 {\displaystyle \nu =1/3} (in terms of hypergeometric functions). The entire class of discrete-stable distributions can be...

all cases), which coincides with the definition before for integer m. Hypergeometric differential equation Cauchy–Euler operator Kreyszig, Erwin (May 10...

In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

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...

predefined set (pathway / NEST), a P-value is computed based on the hypergeometric distribution. It reflects the significance of the observed overlap between...

without replacement, so the correct distribution is the multivariate hypergeometric distribution, but the distributions converge as the population grows...

successes, we may write X ∼ Hypergeometric ⁡ ( N = 8 , K = 4 , n = 4 ) {\displaystyle X\sim \operatorname {Hypergeometric} (N=8,K=4,n=4)} , where N {\displaystyle...

In mathematics, the big q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

theory of orthogonal polynomials is of a definite but limited scope. Hypergeometric series, observed by Felix Klein to be important in astronomy and mathematical...

where ordering does not matter is comparable to a single multivariate hypergeometric distribution. Sampling without replacement where order does matter does...

location missing publisher (link) "The logarithmic solutions of the hypergeometric equation". K. Dan. Vidensk. Selsk. Mat. Fys. SKR. 5: 1–58. 1963. Nörlund–Rice...

h/a;c\\p+q+h/a;\end{bmatrix}},} where 2 F 1 {\displaystyle {}_{2}F_{1}} denotes the hypergeometric series (which converges for all h if c < 1, or for all h / a < q if...

) {\displaystyle _{2}F_{1}(\alpha ,\beta ,\gamma ,z)} is the Gauss hypergeometric function. We may use this fact to bound the second largest eigenvalue...

several closely related identities involving binomial coefficients and hypergeometric functions. Whittaker, E. T. (1936). "Alfred Cardew Dixon. 1865–1936"...

Chu, Wenchang (2007). "Abel's lemma on summation by parts and basic hypergeometric series". Advances in Applied Mathematics. 39 (4): 490–514. doi:10.1016/j...

Lorden, G., & Wang, L. (2022). Optimal and Fast Confidence Intervals for Hypergeometric Successes. Journal of Statistical Planning and Inference, 220, 66-77...

{1}{n}}\right)=2n\int _{0}^{1}{\sqrt[{n}]{1-x^{n}}}\ \mathrm {d} x} Special values of hypergeometric functions at algebraic arguments. 2 F 1 ( − 1 2 , 1 3 ; 4 3 ; − 1 )...

{\displaystyle {}_{2}F_{1}(\ ,\ ;\ ;\ )} is a particular instance of the hypergeometric function. For information on its inverse cumulative distribution function...

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 ) = ( −...

began working on chemically reacting laminar flow, applying Kummer's hypergeometric function to the problem, and control of a stirred tank reactor with...

Narayana numbers multiplied by powers of k. This can be expressed as a hypergeometric function: x n = ∑ i = 1 n N ( n , i ) k i − 1 = ∑ i = 1 n 1 n ( n i...

embedding-based Silhouette community detection can be utilized. For Hypergeometric latent spaces, critical gap method or modified density-based, hierarchical...

polynomials, a class of orthogonal polynomials. They are obtained from the hypergeometric series in certain cases where the series is in fact finite. The Gegenbauer...

of generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have...

The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is The set of constants 0, ∞, 1 in the upper...

}}{2}}\operatorname {erf} (z)} , sin ⁡ ( z ) {\displaystyle \sin(z)} . Hypergeometric functions: For p ∈ N {\displaystyle p\in \mathbb {N} } a general formula...

palindromic string of length p − 1. In 1813 Gauss derived from complex-valued hypergeometric functions what are now called Gauss's continued fractions. They can...

He also put forward a sophisticated Bayesian analysis based on the hypergeometric distribution with a flat prior that could be used to convert trial outcomes...

mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

somewhat less transparent than the earlier proofs, since they rely upon hypergeometric series. See also Particular values of the Riemann zeta function § Odd...

In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

mathematics, the dual q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

Al-Salam–Carlitz polynomials U(a) n(x;q) and V(a) n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Waleed...

the 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 , −...

119–120. doi:10.1093/imamat/34.1.119. Gasper; Rahman (2004). Basic Hypergeometric Series. Cambridge University Press. p. 20–22. ISBN 978-0-521-83357-8...

of two formulas for hypergeometric series, both named after John Dougall: Dougall's formula for the sum of a 7F6 hypergeometric series Dougall's formula...

Series". arXiv:2208.13356 [math.NT]. Zudilin, Wadim (2014-06-01). "Two hypergeometric tales and a new irrationality measure of ζ(2)". Annales mathématiques...

{\mathbf {x} }}\mid {\boldsymbol {\alpha }}')} (Dirichlet-multinomial) Hypergeometric with known total population size, N M (number of target members) Beta-binomial...

ISBN 0-471-54897-9 (Section 6.2.3) Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions", Journal of the Royal Statistical Society, Series B,...

binomial, negative binomial, (discrete) uniform, geometric, Poisson, and hypergeometric. Continuous: (continuous) uniform, exponential, gamma, beta, normal...

inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation...

statistical test applied, the most common being a Fisher's exact test / hypergeometric test. Some methods make use of Bayesian statistics. There is also variability...

"Littlewood polynomials". Analytic Number Theory, Modular Forms and q-Hypergeometric Series. Springer Proceedings in Mathematics & Statistics. Vol. 221....

the real line. The Schwarz triangle function can be given in terms of hypergeometric functions as: s ( α , β , γ ; z ) = z α 2 F 1 ( a ′ , b ′ ; c ′ ; z...

Mostow on the examples of non-arithmetic lattices and monodromy of hypergeometric differential equations in two- and three-dimensional complex hyperbolic...

work has been translated into computer packages that have simplified hypergeometric summation. In 2002, Wilf was awarded the Euler Medal by the Institute...

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...

(jointly with Israel Gelfand and Mikhail Kapranov) of A-systems of hypergeometric equations (also known as GKZ-systems) and development of the theory...

problem Ballistics Airy function Bessel function Legendre polynomials Hypergeometric function Angular velocity Angular momentum Angular acceleration Angular...

Frobenius pseudoprime Frobenius reciprocity Frobenius solution to the hypergeometric equation Frobenius splitting Frobenius theorem (differential topology)...

a scarlet fever he contracted from one of his children. Lauricella hypergeometric series F A , F B , F C , F D {\displaystyle F_{A},F_{B},F_{C},F_{D}}...

-c;N-c;-1/(\lambda h))}}} where 2 F 1 {\displaystyle {}_{2}F_{1}} is the Gaussian hypergeometric function. There are several recursions that can be used to compute P...

of the Dirichlet distribution is a confluent form of the Lauricella hypergeometric series. It is given by Phillips as C F ( s 1 , … , s K − 1 ) = E ⁡ (...

simplification of the main argument.[citation needed] The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of...

existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours...

distribution. In this case, the underlying distribution would be the hypergeometric distribution. The interval boundaries can be computed with numerical...

one that changes the amino acid, a missense mRNA would be detected. A hypergeometric distribution study involving DNA polymerase β replication errors in...

rational function, the power series can be written as a generalized hypergeometric series. The previous example involved an indicial polynomial with a...

112}z^{7}+\cdots \right)} which can be further simplified by the use of hypergeometric series. The power series method can be applied to certain nonlinear...

function (by starting from known special-case solutions of the Gauss hypergeometric differential equation). Also, approximation (11) has been found only...

finite support Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot...

nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion...

analytic. Many special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The...

They also have numerous applications with regard to the summation of hypergeometric series, summations involving the inverse of the central binomial coefficient...

Retrieved 2024-10-02. Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067...

where M ( a , b , z ) {\displaystyle M(a,b,z)} is Kummer's confluent hypergeometric function. The characteristic function is given by: φ ( t ; k ) = M (...

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 , … , 2...

inequality Fisher's linear discriminator Fisher's method Fisher's noncentral hypergeometric distribution Fisher's z-distribution Fisher–Tippett distribution – redirects...

analogue of the Hyperexponential distribution, but it is not called the Hypergeometric distribution, since that name is in use for an entirely different type...

deviation of 1. R has a known density that can be expressed as a confluent hypergeometric function. The distribution of the reciprocal of a t distributed random...

classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation. Let g ^ k {\displaystyle {\hat {\mathfrak {g}}}_{k}}...

applying parabolic coordinates leading to solutions in terms of confluent hypergeometric functions. The broadly applied workaround for the divergence of the...

In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson...

+1;\alpha ,\alpha +\beta +1;{\frac {\lambda }{2}}\right)} (see Confluent hypergeometric function) Variance (type I) e − λ 2 Γ ( α + 2 ) Γ ( α ) Γ ( α + β )...

b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!} is a Hypergeometric function. The Gamma/Gompertz distribution is a flexible distribution...

F5 algorithm) Gosper's algorithm: find sums of hypergeometric terms that are themselves hypergeometric terms Knuth–Bendix completion algorithm: for rewriting...

mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

{\displaystyle \tan x} is irrational. Laczkovich's proof is about the hypergeometric function. In fact, f k ( x ) = 0 F 1 ( k − x 2 ) {\displaystyle...

 380), which also gives the corresponding degenerations of the Gauss hypergeometric function (see Clarkson (2006), p. 372) The Painlevé equations can all...

papers so far related to special functions, modular equation, Basic hypergeometric series and integer partitions.[better source needed] He has so far guided...

Celine Fasenmyer (Ph.D. 1946), mathematician noted for her work on hypergeometric functions and linear algebra Walter Feit (Ph.D. 1955), winner of the...

{\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...

the lituus spiral can be represented as a formula using the Gaussian hypergeometric function: L = 2 θ ⋅ 2 F 1 ⁡ ( − 1 2 , − 1 4 ; 3 4 ; − 1 4 θ 2 ) − 2...

expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex...

a , b ; c ; z ) {\displaystyle _{2}F_{1}(a,b;c;z)} is the Gaussian hypergeometric function and ℜ { z } {\displaystyle \Re \{z\}} denotes the real part...

Dover. Fog, A. (2008), "Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution", Communications in Statistics, Simulation and Computation...

metal band Twilightning Wallenius' noncentral hypergeometric distribution, generalization of the hypergeometric distribution where items are sampled with...

"A determination of the number of real and imaginary roots of the hypergeometric series". Trans. Amer. Math. Soc. 3 (1): 110–131. doi:10.1090/s0002-9947-1902-1500590-4...

finite support Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot...

formulated in terms of a "noncentrality parameter": see noncentral hypergeometric distributions, for example. The noncentrality parameter of the t-distribution...

distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution. Bayesian programming...

different areas; he codified some of the relations between different hypergeometric series, known as contiguity relations. The Kummer surface results from...

mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

function of order α. Using generalized Laguerre polynomials and confluent hypergeometric functions, they can be written as lim n → ∞ n − α L n ( α ) ( z 2 4...

discoveries in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory. 1919 — Viggo Brun defines Brun's constant...

related to the Gamma function Meijer G-function, a generalization of the hypergeometric function Siegel G-function, a class of functions in transcendence theory...

degenerate) obeys a differential equation whose solutions are generalized hypergeometric functions of type N F N − 1 {\displaystyle {}_{N}F_{N-1}} . W-minimal...

Karr's summation in finite terms find closed-form indefinite sums of hypergeometric terms and are implemented in computer algebra systems. The Gregory-Laplace...

(x)P_{n}(\tanh(x)).} where Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by F n ( x ) = 3 F 2 ( − n ,   n + 1 ,   1...

a ) = 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...

the chromosome positions. The genome is divided into intervals and hypergeometric tests are used to identify intervals enriched with mapped reads. DSBs...

after Ronald Fisher: Behrens–Fisher distribution Fisher's noncentral hypergeometric distribution Fisher's z-distribution Fisher's fiducial distribution...

Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer...

{J}}_{0}\left(k{r_{12} \over r_{B}}\right)} where M {\displaystyle M} is a confluent hypergeometric function and J 0 {\displaystyle {\mathcal {J}}_{0}} is a Bessel function...

In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions...

{t}{s}}\right).} Where 2 F 1 {\displaystyle _{2}F_{1}} is the Euler hypergeometric integral. Say we want to simulate an fBm at points 0 = t 0 < t 1 < ⋯...

{\tfrac {1}{2}}\chi ^{2})}}} where M(·,·,·) is the Kummer's confluent hypergeometric function.[circular reference] The variance is: σ 2 = c 2 ( χ 2 ) p +...

integral in question. When 0 < a < b {\displaystyle 0<a<b} , the confluent hypergeometric function of the first kind has the integral representation 1 F 1 ( a...

integral of Eq.19 can be expressed analytically in terms of the Gauss hypergeometric function 2 F 1 ( a , b , c ; w ) {\displaystyle _{2}F_{1}(a,b,c;w)}...

Leonhard Euler, the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series. For real and complex values of z: ∫ arcsin ⁡ ( z ) d z = z arcsin...

n(x|q), introduced by Askey & Wilson (1985), are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter...

"Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories". Nucl. Phys. B. 818 (3): 137–178. arXiv:0801...

functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generalizations, Legendre and Jacobi orthogonal polynomials...

(editors: Tom H. Koornwinder, Jasper V. Stokman) Volume 3: Hypergeometric and Basic Hypergeometric Functions (editor: Mourad Ismail) Further volumes were...

complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function. This was shown in Springer.. A transformation...

1,-{k^{2} \over 4a^{2}}\right).} Here, M is a confluent hypergeometric function. For an application of this integral see Charge density spread...

c ; z ) {\displaystyle {}_{2}F_{1}(a,b;c;z)} is the hypergeometric function. As the hypergeometric function is defined for |z| < 1 but x is unbounded,...

is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be expressed algebraically. More precisely, it is a listing...

In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. The different sine integral definitions...

(2004) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series. Turán's inequalities Askey, Richard; Gasper, George (1976),...

the Al-Salam–Chihara polynomials Qn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam...

the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function. The Rogers–Ramanujan continued fraction is then R ( q ) =...

Struve Functions and Related Functions Confluent Hypergeometric Functions Coulomb Wave Functions Hypergeometric Functions Jacobian Elliptic Functions and Theta...

} 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...

Schilling's theory was presented by Felix Klein in his lectures on hypergeometric functions. Schilling also did research on Reuleaux tetrahedra. He took...

connection between the Clebsch-Gordan Coefficients (CGC) and the Gauss hypergeometric function which was eventually identified as the generating function...

the number 15 in hexadecimal and higher positional systems pFq, the hypergeometric function F-distribution, a continuous probability distribution F-test...

polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by C n ( x ; β | q ) = ( β ; q ) n ( q ; q ) n e i n θ 2 ϕ 1...

,\beta )}}} where 2 F 1 {\displaystyle {}_{2}F_{1}} is the Gauss's hypergeometric function 2F1 . The beta prime distribution may also be reparameterized...

univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution...

and 2 F 1 ( a , b , c , d ) {\displaystyle _{2}F_{1}(a,b,c,d)} is the hypergeometric function. This was derived by using the martingale property of h ( x...

that the number of Gessel excursions with 2n steps admit a simple hypergeometric closed form. This closed form counting function equation became known...

1 ( a ; b ; z ) {\displaystyle {}_{1}F_{1}(a;b;z)} is a generalized hypergeometric series and θ n ( x ; α , β ) {\displaystyle \theta _{n}(x;\alpha ,\beta...

the Γ {\displaystyle \Gamma } -function, it reduces to a generalized hypergeometric series … = ∑ k ≥ 0 ( − 4 ) k ( − 1 / 2 ) k k ! = 1 F 0 ( − 1 / 2 ; ;...

MAA Mary Celine Fasenmyer (1906–1996), Catholic nun whose research on hypergeometric functions prefigured WZ theory Heike Fassbender, German expert in numerical...

British mathematician Lucy Joan Slater published two books about the hypergeometric functions from the Cambridge University Press. 1960s: American research...

known as the Mellin transform. He studied related gamma functions, hypergeometric functions, Dirichlet series and the Riemann ζ function. He was appointed...

Degree Confluence Project, a web-based volunteer project Confluent hypergeometric function, a mathematical function Confluent, a data streaming software...

Whittaker function or Whittaker integral, in the theory of confluent hypergeometric functions. This makes him also the eponym of the Whittaker model in...

{10-x}{3-y}}}{\binom {10}{3}}}} for 0 ≤ y ≤ min ( 3, x ). It is the hypergeometric distribution H ( x; 3, 7 ), or equivalently, H ( 3; x, 10-x ). The corresponding...

Estrada, Christopher; Marcolli, Matilde (2013). "Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models". International...

)^{-1/k}\right)\right],} where 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function. Alternatively, TVaR α ⁡ ( X ) = − γ − β α c k c + 1 ( ( 1...

\Gamma (\nu +j+1)}}.} Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as: f X ( x ; k , λ ) = e − λ...

superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function", Computer Aided Geometric Design, 29 (7): 510–518, doi:10...

Wright, E. Maitland (1935). "The Asymptotic Expansion of the Generalized Hypergeometric Function". Journal of the London Mathematical Society. s1-10 (4): 286–293...

statistical test producing p-values (Fisher's exact test or the test using hypergeometric distribution). This method identifies FGS by considering their relative...

zeros of the Airy function and U {\displaystyle U} is the confluent hypergeometric function. Janson and Louchard (2007) show that f A + ( x ) ∼ 72 6 π...

Zelevinsky in the 1980s as an offshoot of their work on generalized hypergeometric functions. This led to them writing their textbook in which the hyperdeterminant...

computation, including polynomial identities, trigonometric identities and hypergeometric identities. Theorems in mathematics and theories in science are fundamentally...

1088/0305-4470/21/20/005, ISSN 0305-4470, MR 0966792 Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis...

1063/1.530917. hdl:2066/141604. S2CID 13934946. Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis...

T. (1903). "An expression of certain known functions as generalized hypergeometric functions". Bulletin of the American Mathematical Society. 10 (3): 125–134...

Verification of special functions: Gamma function Elliptic functions Hypergeometric functions Hurwitz zeta function Bessel function Matrix function Verification...

function. A similar method can be applied to Euler's integral for the hypergeometric function to give its analytic continuation. Jordan (1887), pp. 243–244...

Log semiring Citations B. C. Carlson (1966). "Some inequalities for hypergeometric functions". Proc. Amer. Math. Soc. 17: 32–39. doi:10.1090/s0002-9939-1966-0188497-6...

not exponential families are the F-distribution, Cauchy distribution, hypergeometric distribution and logistic distribution. Following are some detailed...

area. Ablowitz-Kaup-Newell-Segur (AKNS) system Clairaut's equation Hypergeometric differential equation Jimbo–Miwa–Ueno isomonodromy equations Painlevé...

-1}} The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function. Let ψ b e t a ( t | α , β ) ↔ Ψ B E T A ( ω | α , β ) {\displaystyle...

(2016). "The Lambert- W step-potential – an exactly solvable confluent hypergeometric potential". Physics Letters A. 380 (5–6): 640–644. arXiv:1509.00846...

light Orbital angular momentum of free electrons Circular polarization Hypergeometric-Gaussian modes Laguerre-Gaussian modes Spin angular momentum of light...

{1}{\sqrt {x}}}\right)}} Let m F n {\displaystyle {}_{m}F_{n}} denote the hypergeometric functions, then: f ( x ; 4 3 , 0 , 1 , 0 ) = 3 5 4 2 5 2 π 1 2 Γ ( 7...

\right),} where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed...

Benjamin/Cummings. ISBN 9780321021731. Chapman, D.G. (1951). Some properties of the hypergeometric distribution with applications to zoological sample censuses. UC Publications...

Margaret Maltby and Grace Chisholm. Her first paper, on the topic of hypergeometric functions, was published in 1894. The Association of Collegiate Alumnae...

ISSN 0950-1207, JSTOR 92601 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed...

Wiktionary, the free dictionary. PFQ or pFq can refer to: Generalized hypergeometric function, a family of mathematical functions denoted as p F q {\displaystyle...

noncentrality parameter μ can be expressed in several forms. The confluent hypergeometric function form of the density function is f ( x ) = Γ ( ν + 1 2 ) ν π...

functions (of any order) can be expressed in terms of the generalized hypergeometric function 1F2: H α ( z ) = z α + 1 2 α π Γ ( α + 3 2 ) 1 F 2 ( 1 ; 3...

contributor to the field of special functions, especially confluent hypergeometric functions. A native of Brooklyn, New York, Miller attended George W...

polynomials pn(x;a|q) or Wall polynomials Wn(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to...

particular integrals of above equation can be expressed in terms of hypergeometric functions. For two-dimensional potential flow, the continuity equation...

(1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by p n ( q − x + q x + 1 c d ; a , b , c , d ; q ) = 4 ϕ 3...

( z ) . {\displaystyle E_{q}(z).} It is a special case of the basic hypergeometric series, E q ( z ) = 1 ϕ 1 ( 0 0 ; z ) = ∑ n = 0 ∞ q ( n 2 ) ( − z )...

worked in the field of theory of analytic functions, in particular hypergeometric functions. Dobrowolski, Wiaczesław A. (1974), "Kazimierz Abramowicz...

complex-valued continued fraction via a clever identity involving the hypergeometric function 1892 Henri Padé defined Padé approximant 1972 Bill Gosper –...

statistically overrepresented terms in the user's list of genes using hypergeometric distribution. MOET also displays the corresponding Bonferroni correction...

recurrence relation for values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure)...

Paper by James D. Thorne with a direct algebraic solution based on hypergeometric series reversion of all hyperbolic and elliptic cases of the Lambert...

Ludwig Siegel. Among these functions are such special functions as the hypergeometric function, cylinder, spherical functions and so on. Using the FEE, it...

215. Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in...

generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:. They also play an important...

ISSN 0950-1207, JSTOR 92601 Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed...

(1:C) Geometric distribution / (1:D) Half circle distribution / (1:C) Hypergeometric distribution / (1:D) Normal distribution / Gau Integration of the normal...

applied in CFA are Pearson's chi-squared test, the binomial test or the hypergeometric test of Lehmacher). If the statistical test suggests for a given α {\displaystyle...

Geometric distribution Negative binomial distribution Poisson distribution Hypergeometric distribution Zeta distribution Uniform distribution (continuous) Normal...

and food consumption Risk compensation Challenging election results (Hypergeometric distribution) Condorcet's jury theorem Cost-benefit analysis of renewable...

mathematician, author of the standard work of choice in the field of Basic Hypergeometric Series S.M. Ullah, soil scientist and environmentalist who researched...

integer is known as the topological charge, or strength, of the vortex. A hypergeometric-Gaussian mode (HyGG) has an optical vortex in its center. The beam,...

function Heilbronn triangle problem Helly family Hypergeometric function identities Hypergeometric series Hypergraph Incidence structure Induction puzzles...

SimRank NASARI: Sparse vector representations constructed by applying the hypergeometric distribution over the Wikipedia corpus in combination with BabelNet...

ISBN 0-387-97509-8, MR 1113282 Karatsuba, E. A.; Vuorinen, M. (2001), "On hypergeometric functions and generalizations of Legendre's relation", J. Math. Anal...

74....1W. Wicksell, S.D. (1917). "XXXIX. The application of solid hypergeometrical series to frequency distributions in space". The London, Edinburgh...

Thomas Jan Stieltjes and Carl Severin Wigert) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function...

a+c;a+b+2c;1-z),\;\;\;0<z<1} where 2 F 1 {\displaystyle {_{2}F_{1}}} is the Gauss hypergeometric function defined by the Euler integral 2 F 1 ( a , b , c , z ) = Γ (...

investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory. Paul Erdős published more papers than...

1 {\displaystyle F_{2,1}} is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of F N...

{}_{2}F_{1}(1-K,1-N;2;2).} where 2 F 1 {\displaystyle {}_{2}F_{1}} is the hypergeometric function. Vector quantization ACELP Opus (audio format) Fischer, Thomas...

rigour precluded further development in calculus. Gauss in his study of hypergeometric series (1813) for the first time rigorously investigated the conditions...

Volume 1 (PDF) (2 ed.). Retrieved April 30, 2025. Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis...

mechanics has been in the area of Nikiforov-Uvarov theory of generalized hypergeometric differential equation, used to solve the Schrödinger equation and to...

}{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),} where pFq is a generalized hypergeometric function. Anger function Lommel polynomial Struve function Weber function...

1103/PhysRev.42.210. Schrödinger, E. (1941). "The Factorization of the Hypergeometric Equation". Proc. Roy. Irish Acad. A. 47: 53–54. JSTOR 20488434. Barut...

theorem (CLT). Special case of distribution parametrization: X is a hypergeometric (m, N, n) random variable. If n and m are large compared to N, and p...

integrals", Q. J. Pure Appl. Math. 41 193–203. Exton, Harold (1983). Q-hypergeometric functions and applications. Chichester [West Sussex]: E. Horwood. ISBN 978-0470274538...

of the algorithm is O(n!). The name for the distribution method is Hypergeometric Distribution (Hoffman). Tajima's Neutrality Test — The purpose of Tajima's...

{x}{n}}=x(x-1)\cdots (x-n+1)} denotes the falling factorial. In terms of the Gaussian hypergeometric function, we have g n ( x ) = x ⋅ 2 F 1 ( 1 − n , 1 − x ; 2 ; 2 ) ....

precise statistics, namely the Fisher-Yates exact test based on the hypergeometric distribution; thus, unlike t-scores, z-scores, chi-square tests etc...

1007/BF02101296. S2CID 119128058. Felder, Giovanni; Varchenko, Alexander (2004). "Hypergeometric theta functions and elliptic Macdonald polynomials". International Mathematics...

generating functions and explicit forms of MRM-triples by means of q-hypergeometric series. Infinite Dimensional Analysis, Quantum Probability and Related...

Reports, vol.32, pp. 16–43, 96-177, 1977 ISSN 0031-7918. "The Gauss hypergeometric ratio as a positive real function", SIAM Journal on Mathematical Analysis...

integral. Andrews, George E. (1975), "Problems and prospects for basic hypergeometric functions", Theory and application of special functions (Proc. Advanced...

then the corresponding conformal blocks can be written in terms of the hypergeometric function. As first explained by Witten, the space of conformal blocks...

Examples of such conditional distributions are the normal, binomial, beta, hypergeometric and geometric distributions, which are not all NEF-QVF. NEF-QVF have...

interest in computation in general, particularly in Bessel and the hypergeometric functions. He spent the rest of his life developing methods to compute...

(x^{n})_{n\in \mathbb {N} }} ). Other algorithms which compute rational or hypergeometric solutions of a linear recurrence equation with polynomial coefficients...

)^{2}}}t^{r}\,_{2}F_{2}\left(1,1;r+1,r+1;-t\right)\right),} where 2F2 is a hypergeometric function. The r=1 case for the harmonic numbers is a classical result...

St. P. Richards: Gross, Kenneth I.; Richards, Donald St. P. (1991). "Hypergeometric functions on complex matrix space". Bull. Amer. Math. Soc. (N.S.). 24...

mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by Heatley (1943), Weisstein, as T ( m , n , r ) =...

{J_{s}(z{\sqrt {1-x^{2}}})}{{\sqrt {1-x^{2}}}^{s}}}\,dx,} the confluent hypergeometric function M ( a , s , z ) = Γ ( s ) ∑ k = 0 ∞ ( − 1 t ) k L k ( − a −...

of clusters (with a fixed number of set elements N). By adopting a hypergeometric model of randomness, it can be shown that the expected mutual information...

the classical orthogonal polynomials. Appell sequence Askey scheme of hypergeometric orthogonal polynomials Polynomial sequences of binomial type Biorthogonal...