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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...

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

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

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

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

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

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

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

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

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

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

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...

{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...

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...

{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...

elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series...

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...

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

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...

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...

{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...

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

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

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

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

–2t/(1–t2) An explicit expression for them in terms of the generalized hypergeometric function 3F0: s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 −...

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...

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 −...

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

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

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

} 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 ∞...

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

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

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

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

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

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

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

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

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

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

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...

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

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

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 −...

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...

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

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

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

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)...

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...

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

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

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...

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

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

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

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

{\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...

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...

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

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

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

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...

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...

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

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

ordinary differential equation of hypergeometric type, whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's...

}&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...

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...

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

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

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...

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

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

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

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...

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

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

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

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

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

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

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

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

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

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

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

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...

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...

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

{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 ...

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...

\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...

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

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

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

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

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...

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

{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...

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

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

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

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

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...

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

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

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

_{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...

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...

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

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...

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

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...

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

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

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

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

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...

{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...

{\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...

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...

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 ⁡...

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

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

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

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

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...

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

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 ] =...

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

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

{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...

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

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...

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...

} 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...

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

\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...

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

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...

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

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

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...

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...

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

)^{-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 −...

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

{\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...

{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...

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

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...

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...

; 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...

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

In mathematics, Lentz's algorithm is an algorithm to evaluate continued fractions,[full citation needed] and was originally devised to compute tables of...

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...

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...

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

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

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

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

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

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

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...

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

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

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

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...

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

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

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

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...

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

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

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...

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 , −...

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

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

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...

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...

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

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

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,...

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

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...

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

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

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

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

}{(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 ,...

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

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

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

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...

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

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

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

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

(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...

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

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

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

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

(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...

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 ⁡ (...

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...

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

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

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

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...

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...

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...

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

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

}{\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...

first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions. It also holds that e π 2 + e − π 2 π e γ = ∏ n = 1 ∞ ( e...

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

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

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

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

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

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

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

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

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...

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

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 (...

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

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

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

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):...

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

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

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

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...

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

{\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...

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

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

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...

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

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 (...

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

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

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

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

)^{-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...

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

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

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

{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...

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

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

( 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 )...

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

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...

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

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

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

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

{\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...

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...

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

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

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

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

_{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...

)}}\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...

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,...

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

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

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

waterfall 1949 Estimation in probability and its application 1952 Class of hypergeometric functions 1953a Planning of electric energy production 1953b Statistical...

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

\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...

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

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}}...

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

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...

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...

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

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...

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

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...

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

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

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...

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...

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

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...

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

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

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...

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...

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

{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 )...

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...

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

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

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

} 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...

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

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

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

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

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

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...

-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...

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

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

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 −...

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)...

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

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...

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 ) = ( −...

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...

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

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

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,...

-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...

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

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

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...

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

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

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...

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...

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

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

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

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

\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...

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...

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

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

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

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

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

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

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

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 (...

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

+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 )...

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

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

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)}...

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

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...

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

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...

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

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

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

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 ) =...

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

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

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

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...

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 ; ;...

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,...

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...

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

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

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...

{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...

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...

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

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

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

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...

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

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

"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...

{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...

{}_{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...

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

{r_{12}}{r_{B}}}\right)}} where M {\displaystyle M} is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we...

{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 < ⋯...

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

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

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...

{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...

(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...

n(x|q), introduced by Askey & Wilson (1985), 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...

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

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...

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

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

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

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

operator/connective, the analytical result is not always a Beta PDF and can involve hypergeometric series. In such cases, subjective logic always approximates the result...

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

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

Accurate Tandem Mass Spectral Peptide Identification by Multivariate Hypergeometric Analysis". Journal of Proteome Research. 6 (2): 654–61. doi:10.1021/pr0604054...

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

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

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

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...

(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...

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

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...

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...

"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...

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

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...

(2011-04-13) [2010-12-23]. "The integrals in Gradshteyn and Ryzhik. Part 20: Hypergeometric functions" (PDF). Scientia. Series A: Mathematical Sciences. 21 (published...

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

{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 ) ....

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...

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

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...

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

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

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

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

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

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

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...

{\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 +...

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...

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

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

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

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

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

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

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...

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

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

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

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

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...

S2CID 119599225. Rains, Eric M. (2010). "Transformations of elliptic hypergeometric integrals" (PDF). Annals of Mathematics. 171 (1): 169–243. doi:10.4007/annals...

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...

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

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

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

classical spectral theory of ordinary differential equations applied to the hypergeometric equation (Mehler, Weyl, Fock); variants of Hadamard's method of descent...

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

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

1134/S1995080220100066. S2CID 229510108. Brychkov Yu.A., Savischenko N.V. (2021). "Hypergeometric Functions of Several Variables and Evaluation of Error Probability in...

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

complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function. This was shown in Springer 1979 problem...

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

{3}{2}},2;-z^{2}\right),} where 2 F 2 {\displaystyle {}_{2}F_{2}} is a hypergeometric function. In order for the function to approach zero as x approaches...

are made without replacement, the distribution follows a multivariate hypergeometric distribution. Once again, let α 0 = ∑ α k {\displaystyle \alpha _{0}=\sum...

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...

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

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

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

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

Lecture Series), AMS 1992, ISBN 0821870025 Varchenko, A. Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups...

 In particular, he related these sums to certain classical confluent hypergeometric differential equations. This relationship generalized the study due...

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

contributor to the field of special functions, especially confluent hypergeometric functions Teri Perl (B.A. 1947), mathematics educator, co-founder of...

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...

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

1936. The work of Hirsch is primarily on differential equations and hypergeometric functions. He published seven papers about it in Mathematische Annalen...

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

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

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...

introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case p > q + 1. The underlying objective was to...

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...

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...

Affymetrix chips, the model-based analysis of tiling array (MAT) or hypergeometric analysis of tiling-arrays (HAT) are effective peak-seeking algorithms...

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

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

algorithms Chudnovsky algorithm — fast algorithm that calculates a hypergeometric series Bailey–Borwein–Plouffe formula — can be used to compute individual...

physics, Baer functions cannot in general be expressed in terms of hypergeometric functions. The Baer wave equation is a generalization which results...

, b ; z ) {\displaystyle _{1}F_{1}(a,b;z)} is the Kummer confluent hypergeometric function. This function has usually very good convergence properties...

not periodic, and it cannot be written in terms of sinusoidal or even hypergeometric functions. (See Mathieu function for more about the Mathieu cosine function...

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

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...

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

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

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

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

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

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

)^{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...

however, obtain the following formula in terms of the generalized hypergeometric function: F ( n ) = P ( X ≤ n ) = 1 − 1 F ν − 1 ( ; n + 2 , … , n +...

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

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...

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...

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...

MR 1217488. Orlik, Peter; Terao, Hiroaki (2007) [2001]. Arrangements and hypergeometric integrals. MSJ Memoirs. Vol. 9 (2nd ed.). Tokyo: Mathematical Society...

case when s = 0 or s = n can be dealt with, we first go back to the hypergeometric distribution, denoted by H y p ( s | N , n , S ) {\displaystyle \mathrm...