Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function ₂F₁(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

The term "hypergeometric function" sometimes refers to the generalized hypergeometric function. For other hypergeometric functions see See also.

For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.

History[edit]

The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.

Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813).

Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential equation it satisfies.

Riemann showed that the second-order differential equation for ₂F₁(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.

The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).

Integral formulas[edit]

Euler type[edit]

If B is the beta function then

a 2-variable generalization of hypergeometric series

Appell series

where the ratio of terms is a periodic function of the index

Basic hypergeometric series

_pH_p are similar to generalized hypergeometric series, but summed over all integers

Bilateral hypergeometric series

₁F₀

Binomial series

₁F₁(a;c;z)

Confluent hypergeometric series

where the ratio of terms is an elliptic function of the index

Elliptic hypergeometric series

an integral representation of ₂F₁

Euler hypergeometric integral

an extension of the Meijer G-function

Fox H-function

a generalization of the generalized hypergeometric function

Fox–Wright function

Frobenius solution to the hypergeometric equation

introduced by I. M. Gelfand.

General hypergeometric function

_pF_q where the ratio of terms is a rational function of the index

Generalized hypergeometric series

where the ratio of terms is a constant

Geometric series

solutions of second order ODE's with four regular singular points

Heun function

34 distinct convergent hypergeometric series in two variables

Horn function

7 hypergeometric functions of 2 variables

Humbert series

a discrete probability distribution

Hypergeometric distribution

the multivariate generalization of the hypergeometric series

Hypergeometric function of a matrix argument

hypergeometric series of two variables

Kampé de Fériet function

hypergeometric series of three variables

Lauricella hypergeometric series

an extension of the generalized hypergeometric series _pF_q to the case p>q+1.

MacRobert E-function

an extension of the generalized hypergeometric series _pF_q to the case p>q+1.

Meijer G-function

a terminating form of the elliptic hypergeometric series

Modular hypergeometric series

a special sort of elliptic hypergeometric series.

Theta hypergeometric series

special functions in two-dimensional conformal field theory that reduce to hypergeometric functions in some cases.

Virasoro conformal blocks

; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR 1688958.

Andrews, George E.

Bailey, W.N. (1935). (PDF). Cambridge University Press. Archived from the original (PDF) on 2017-06-24. Retrieved 2016-07-23.

Generalized Hypergeometric Series

(2002), Gauss' hypergeometric function. (lecture notes reviewing basics, as well as triangle maps and monodromy)

Beukers, Frits

Olde Daalhuis, Adri B. (2010), , in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

"Hypergeometric function"

; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions (PDF). Vol. I. New York – Toronto – London: McGraw–Hill Book Company, Inc. ISBN 978-0-89874-206-0. MR 0058756.

Erdélyi, Arthur

Gasper, George & (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.

Rahman, Mizan

(1813). "Disquisitiones generales circa seriem infinitam $1+{\tfrac {\alpha \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}~x~x+{\mbox{etc.}}$ ". Commentationes Societatis Regiae Scientarum Gottingensis Recentiores (in Latin). 2. Göttingen.

Gauss, Carl Friedrich

Gelfand, I. M.; Gindikin, S.G. & Graev, M.I. (2003) [2000]. . Translations of Mathematical Monographs. Vol. 220. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2932-5. MR 2000133.

Selected topics in integral geometry

Gessel, Ira & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal on Mathematical Analysis. 13 (2): 295–308. :10.1137/0513021. ISSN 0036-1410. MR 0647127.

doi

(1881). "Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique". Annales Scientifiques de l'École Normale Supérieure (in French). 10: 3–142. doi:10.24033/asens.207. Retrieved 2008-10-16.

Goursat, Édouard

Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. 0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)

ISBN

Hille, Einar (1976). . Dover. ISBN 0-486-69620-0.

Ordinary differential equations in the complex domain

(1944). Ordinary Differential Equations. Dover Publications.

Ince, E. L.

Klein, Felix (1981). . Grundlehren der Mathematischen Wissenschaften (in German). Vol. 39. Berlin, New York: Springer-Verlag. ISBN 978-3-540-10455-1. MR 0668700.

Vorlesungen über die hypergeometrische Funktion

Koepf, Wolfram (1995). . Journal of Symbolic Computation. 20 (4): 399–417. doi:10.1006/jsco.1995.1056. ISSN 0747-7171. MR 1384455.

"Algorithms for m-fold hypergeometric summation"

Kummer, Ernst Eduard (1836). . Journal für die reine und angewandte Mathematik (in German). 15: 39–83, 127–172. ISSN 0075-4102.

"Über die hypergeometrische Reihe $1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+\cdots$ "

Lavoie, J. L.; Grondin, F.; Rathie, A.K. (1996). . J. Comput. Appl. Math. 72 (2): 293–300. doi:10.1016/0377-0427(95)00279-0.

"Generalizations of Whipple's theorem on the sum of a ₃F₂"

Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). . Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

"Section 6.13. Hypergeometric Functions"

Rakha, M.A.; Rathie, Arjun K. (2011). . Bull. Korean Math. Soc. 48 (1): 151–156. doi:10.4134/BKMS.2011.48.1.151.

"Extensions of Euler's type-II transformation and Saalschutz's theorem"

Rathie, Arjun K.; Paris, R.B. (2007). "An extension of the Euler's-type transformation for the 3F2 series". Far East J. Math. Sci. 27 (1): 43–48.

(1857). "Beiträge zur Theorie der durch die Gauss'sche Reihe F(α, β, γ, x) darstellbaren Functionen". Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen (in German). 7. Göttingen: Verlag der Dieterichschen Buchhandlung: 3–22. (a reprint of this paper can be found in "All publications of Riemann" (PDF).)

Riemann, Bernhard

(1960). Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press. MR 0107026.

Slater, Lucy Joan

(1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)

Slater, Lucy Joan

Vidunas, Raimundas (2005). "Transformations of some Gauss hypergeometric functions". Journal of Symbolic Computation. 178 (1–2): 473–487. :math/0310436. Bibcode:2005JCoAM.178..473V. doi:10.1016/j.cam.2004.09.053. S2CID 119596800.

arXiv

Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.

& Watson, G.N. (1927). A Course of Modern Analysis. Cambridge, UK: Cambridge University Press.

Whittaker, E.T.

Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig – Wiesbaden: Friedr. Vieweg & Sohn. 3-528-06925-2. MR 1453580.

ISBN

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Hypergeometric function"

John Pearson, (University of Oxford, MSc Thesis)

Computation of Hypergeometric Functions

Marko Petkovsek, Herbert Wilf and Doron Zeilberger, (freely downloadable)

The book "A = B"

"Hypergeometric Function". MathWorld.

Hypergeometric function

History[edit]

Integral formulas[edit]

Euler type[edit]

Appell series

Basic hypergeometric series

Bilateral hypergeometric series

Binomial series

Confluent hypergeometric series

Elliptic hypergeometric series

Euler hypergeometric integral

Fox H-function

Fox–Wright function

Frobenius solution to the hypergeometric equation

General hypergeometric function

Generalized hypergeometric series

Geometric series

Heun function

Horn function

Humbert series

Hypergeometric distribution

Hypergeometric function of a matrix argument

Kampé de Fériet function

Lauricella hypergeometric series

MacRobert E-function

Meijer G-function

Modular hypergeometric series

Theta hypergeometric series

Virasoro conformal blocks

Andrews, George E.

Generalized Hypergeometric Series

Beukers, Frits

"Hypergeometric function"

Erdélyi, Arthur

Rahman, Mizan

Gauss, Carl Friedrich

Selected topics in integral geometry

doi

Goursat, Édouard

ISBN

Ordinary differential equations in the complex domain

Ince, E. L.

Vorlesungen über die hypergeometrische Funktion

"Algorithms for m-fold hypergeometric summation"

"Generalizations of Whipple's theorem on the sum of a 3F2"

"Section 6.13. Hypergeometric Functions"

"Extensions of Euler's type-II transformation and Saalschutz's theorem"

Riemann, Bernhard

Slater, Lucy Joan

Slater, Lucy Joan

arXiv

Whittaker, E.T.

ISBN

"Hypergeometric function"

Computation of Hypergeometric Functions

The book "A = B"

Weisstein, Eric W.

"Generalizations of Whipple's theorem on the sum of a ₃F₂"