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.
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
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).
The hypergeometric series
The hypergeometric function is defined for |z| < 1 by the power series
It is undefined (or infinite) if c equals a non-positive integer. Here (q)n is the (rising) Pochhammer symbol, which is defined by:
The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial:
For complex arguments z with |z| ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line z ≥ 1.
As c → −m, where m is a non-negative integer, one has 2F1(z) → ∞. Dividing by the value Γ(c) of the gamma function, we have the limit:
2F1(z) is the most common type of generalized hypergeometric series pFq, and is often designated simply F(z).
Differentiation formulas
Using the identity , it is shown that
and more generally,
Special cases
Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are
When a=1 and b=c, the series reduces into a plain geometric series, i.e.
hence, the name hypergeometric. This function can be considered as a generalization of the geometric series.
The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
Legendre functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example
Several orthogonal polynomials, including Jacobi polynomials P(α,β)
n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using
Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.
Given , let
Then
is the modular lambda function, where
The j-invariant, a modular function, is a rational function in .
Incomplete beta functions Bx(p,q) are related by
The complete elliptic integrals K and E are given by
The hypergeometric differential equation
The hypergeometric function is a solution of Euler's hypergeometric differential equation
which has three regular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order linear differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.
Solutions at the singular points
Solutions to the hypergeometric differential equation are built out of the hypergeometric series 2F1(a,b;c;z). The equation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at a regular singular point. This gives 3 × 2 = 6 special solutions, as follows.
Around the point z = 0, two independent solutions are, if c is not a non-positive integer,
and, on condition that c is not an integer,
If c is a non-positive integer 1−m, then the first of these solutions does not exist and must be replaced by The second solution does not exist when c is an integer greater than 1, and is equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a more complicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plus another series in powers of z, involving the digamma function. See Olde Daalhuis (2010) for details.
Around z = 1, if c − a − b is not an integer, one has two independent solutions
and
Around z = ∞, if a − b is not an integer, one has two independent solutions
and
Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated.
Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (6
3) = 20 linear relations between them called connection formulas.
Kummer's 24 solutions
A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group W(Dn) of order 2n−1n!. The hypergeometric equation is the case n = 3, with group of order 24 isomorphic to the symmetric group on 4 points, as first described by Kummer. The appearance of the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a,b;c;z) to one of
which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independent solution to the differential equation.)
Applying Kummer's 24 = 6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities
Q-form
The hypergeometric differential equation may be brought into the Q-form
by making the substitution u = wv and eliminating the first-derivative term. One finds that
and v is given by the solution to
which is
The Q-form is significant in its relation to the Schwarzian derivative (Hille 1976, pp. 307–401).
Schwarz triangle maps
The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.
where k is one of the points 0, 1, ∞. The notation
is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.
Note that each triangle map is regular at z ∈ {0, 1, ∞} respectively, with
and
In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a generalization of the Schwarz–Christoffel mapping to triangles with circular arcs. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.
Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether λ + μ + ν – 1 is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r).
Monodromy group
The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point. That is, when the path winds around a singularity of 2F1, the value of the solutions at the endpoint will differ from the starting point.
Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):
where π1 is the fundamental group. In other words, the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices. The monodromy representation of the fundamental group can be computed explicitly in terms of the exponents at the singular points.[1] If (α, α'), (β, β') and (γ,γ') are the exponents at 0, 1 and ∞, then, taking z0 near 0, the loops around 0 and 1 have monodromy matrices
where
If 1−a, c−a−b, a−b are non-integer rational numbers with denominators k,l,m then the monodromy group is finite if and only if , see Schwarz's list or Kovacic's algorithm.
Integral formulas
Euler type
If B is the beta function then
provided that z is not a real number such that it is greater than or equal to 1. This can be proved by expanding (1 − zx)−a using the binomial theorem and then integrating term by term for z with absolute value smaller than 1, and by analytic continuation elsewhere. When z is a real number greater than or equal to 1, analytic continuation must be used, because (1 − zx) is zero at some point in the support of the integral, so the value of the integral may be ill-defined. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations.
Other representations, corresponding to other branches, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy action.
Barnes integral
Barnes used the theory of residues to evaluate the Barnes integral
as
where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a − 1, ..., −b, −b − 1, ... . This is valid as long as z is not a nonnegative real number.
John transform
The Gauss hypergeometric function can be written as a John transform (Gelfand, Gindikin & Graev 2003, 2.1.2).
Gauss' contiguous relations
The six functions
are called contiguous to 2F1(a, b; c; z). Gauss showed that 2F1(a, b; c; z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, c, and z. This gives
relations, given by identifying any two lines on the right hand side of
where F = 2F1(a, b; c; z), F(a+) = 2F1(a + 1, b; c; z), and so on. Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form
where m, n, and l are integers.
Gauss' continued fraction
Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:
Transformation formulas
Transformation formulas relate two hypergeometric functions at different values of the argument z.
Fractional linear transformations
Euler's transformation is
It follows by combining the two Pfaff transformations
which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see Rathie & Paris (2007) and Rakha & Rathie (2011). It can also be written as linear combination
Quadratic transformations
If two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by Kummer (1836), and a complete list was given by Goursat (1881). A typical example is
Higher order transformations
If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by Goursat (1881). A typical example is
There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers (Vidunas 2005). For example,
Values at special points z
See Slater (1966, Appendix III) for a list of summation formulas at special points, most of which also appear in Bailey (1935). Gessel & Stanton (1982) gives further evaluations at more points. Koepf (1995) shows how most of these identities can be verified by computer algorithms.
Special values at z = 1
Gauss's summation theorem, named for Carl Friedrich Gauss, is the identity
which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity as a special case.
For the special case where ,
Dougall's formula generalizes this to the bilateral hypergeometric series at z = 1.
Kummer's theorem (z = −1)
There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformation to change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer:
which follows from Kummer's quadratic transformations
and Gauss's theorem by putting z = −1 in the first identity. For generalization of Kummer's summation, see Lavoie, Grondin & Rathie (1996).
Values at z = 1/2
Gauss's second summation theorem is
Bailey's theorem is
For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see Lavoie, Grondin & Rathie (1996).
Other points
There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in Gessel & Stanton (1982) and Koepf (1995). Some typical examples are given by
which can be restated as
whenever −π < x < π and T is the (generalized) Chebyshev polynomial.
See also
- Appell series, a 2-variable generalization of hypergeometric series
- Basic hypergeometric series where the ratio of terms is a periodic function of the index
- Bilateral hypergeometric series pHp are similar to generalized hypergeometric series, but summed over all integers
- Binomial series 1F0
- Confluent hypergeometric series 1F1(a;c;z)
- Elliptic hypergeometric series where the ratio of terms is an elliptic function of the index
- Euler hypergeometric integral, an integral representation of 2F1
- Fox H-function, an extension of the Meijer G-function
- Fox–Wright function, a generalization of the generalized hypergeometric function
- Frobenius solution to the hypergeometric equation
- General hypergeometric function introduced by I. M. Gelfand.
- Generalized hypergeometric series pFq where the ratio of terms is a rational function of the index
- Geometric series, where the ratio of terms is a constant
- Heun function, solutions of second order ODE's with four regular singular points
- Horn function, 34 distinct convergent hypergeometric series in two variables
- Humbert series 7 hypergeometric functions of 2 variables
- Hypergeometric distribution, a discrete probability distribution
- Hypergeometric function of a matrix argument, the multivariate generalization of the hypergeometric series
- Kampé de Fériet function, hypergeometric series of two variables
- Lauricella hypergeometric series, hypergeometric series of three variables
- MacRobert E-function, an extension of the generalized hypergeometric series pFq to the case p>q+1.
- Meijer G-function, an extension of the generalized hypergeometric series pFq to the case p>q+1.
- Modular hypergeometric series, a terminating form of the elliptic hypergeometric series
- Theta hypergeometric series, a special sort of elliptic hypergeometric series.
- Virasoro conformal blocks, special functions in two-dimensional conformal field theory that reduce to hypergeometric functions in some cases.
References
- Andrews, George E.; 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.
- Bailey, W.N. (1935). Generalized Hypergeometric Series (PDF). Cambridge University Press. Archived from the original (PDF) on 2017-06-24. Retrieved 2016-07-23.
- Beukers, Frits (2002), Gauss' hypergeometric function. (lecture notes reviewing basics, as well as triangle maps and monodromy)
- Olde Daalhuis, Adri B. (2010), "Hypergeometric function", 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.
- Erdélyi, Arthur; 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.
- Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
- Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriem infinitam ". Commentationes Societatis Regiae Scientarum Gottingensis Recentiores (in Latin). Göttingen. 2.
- Gelfand, I. M.; Gindikin, S.G. & Graev, M.I. (2003) [2000]. Selected topics in integral geometry. Translations of Mathematical Monographs. Vol. 220. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2932-5. MR 2000133.
- Gessel, Ira & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal on Mathematical Analysis. 13 (2): 295–308. doi:10.1137/0513021. ISSN 0036-1410. MR 0647127.
- Goursat, Édouard (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.
- Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. ISBN 0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)
- Hille, Einar (1976). Ordinary differential equations in the complex domain. Dover. ISBN 0-486-69620-0.
- Ince, E. L. (1944). Ordinary Differential Equations. Dover Publications.
- Klein, Felix (1981). Vorlesungen über die hypergeometrische Funktion. Grundlehren der Mathematischen Wissenschaften (in German). Vol. 39. Berlin, New York: Springer-Verlag. ISBN 978-3-540-10455-1. MR 0668700.
- Koepf, Wolfram (1995). "Algorithms for m-fold hypergeometric summation". Journal of Symbolic Computation. 20 (4): 399–417. doi:10.1006/jsco.1995.1056. ISSN 0747-7171. MR 1384455.
- Kummer, Ernst Eduard (1836). "Über die hypergeometrische Reihe ". Journal für die reine und angewandte Mathematik (in German). 15: 39–83, 127–172. ISSN 0075-4102.
- Lavoie, J. L.; Grondin, F.; Rathie, A.K. (1996). "Generalizations of Whipple's theorem on the sum of a 3F2". J. Comput. Appl. Math. 72 (2): 293–300. doi:10.1016/0377-0427(95)00279-0.
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). "Section 6.13. Hypergeometric Functions". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
- Rakha, M.A.; Rathie, Arjun K. (2011). "Extensions of Euler's type-II transformation and Saalschutz's theorem". Bull. Korean Math. Soc. 48 (1): 151–156. doi:10.4134/BKMS.2011.48.1.151.
- 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.
- Riemann, Bernhard (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). Göttingen: Verlag der Dieterichschen Buchhandlung. 7: 3–22. (a reprint of this paper can be found in "All publications of Riemann" (PDF).)
- Slater, Lucy Joan (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)
- Vidunas, Raimundas (2005). "Transformations of some Gauss hypergeometric functions". Journal of Symbolic Computation. 178 (1–2): 473–487. arXiv:math/0310436. Bibcode:2005JCoAM.178..473V. doi:10.1016/j.cam.2004.09.053. S2CID 119596800.
- Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.
- Whittaker, E.T. & Watson, G.N. (1927). A Course of Modern Analysis. Cambridge, UK: Cambridge University Press.
- Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig – Wiesbaden: Friedr. Vieweg & Sohn. ISBN 3-528-06925-2. MR 1453580.
External links
- "Hypergeometric function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- John Pearson, Computation of Hypergeometric Functions (University of Oxford, MSc Thesis)
- Marko Petkovsek, Herbert Wilf and Doron Zeilberger, The book "A = B" (freely downloadable)
- Weisstein, Eric W. "Hypergeometric Function". MathWorld.