Inmathematics, the Gaussian or ordinaryhypergeometric function₂F₁(a,b;c;z) is aspecial function represented by thehypergeometric series, that includes many other special functions asspecific orlimiting cases. It is a solution of a second-orderlinearordinary differential equation (ODE). Every second-order linear ODE with threeregular singular points can be transformed into this equation.

For systematic lists of some of the many thousands of publishedidentities involving the hypergeometric function, see the reference works byErdélyi et al. (1953) andOlde 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.

The term "hypergeometric series" was first used byJohn Wallis in his 1655 bookArithmetica Infinitorum.

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

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

Riemann showed that the second-order differential equation for₂F₁(z), examined in the complex plane, could be characterised (on theRiemann sphere) by its threeregular singularities.

The cases where the solutions arealgebraic functions were found byHermann Schwarz (Schwarz's list).

The hypergeometric function is defined for|z| < 1 by thepower series

$${\displaystyle {}_2{F}_1(a,b;c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!}=1+\frac{ab}{c}\frac{z}{1!}+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}+\cdots .}$$

It is undefined (or infinite) ifc equals a non-positiveinteger. Here(q)_n is the (rising)Pochhammer symbol,^{[note 1]} which is defined by:

The series terminates if eithera orb is a nonpositive integer, in which case the function reduces to a polynomial:

$${\displaystyle {}_2{F}_1(-m,b;c;z)=\sum _{n=0}^m(-1{)}^n(\genfrac{}{}{0ex}{}{m}{n})\frac{(b{)}_n}{(c{)}_n}{z}^n.}$$

For complex argumentsz with|z| ≥ 1 it can beanalytically 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 linez ≥ 1.

Asc → −m, wherem is a non-negative integer, one has₂F₁(z) → ∞. Dividing by the valueΓ(c) of thegamma function, we have the limit:

$${\displaystyle \underset{c\to -m}{lim}\frac{{}_2{F}_1(a,b;c;z)}{\mathrm{\Gamma }(c)}=\frac{(a{)}_{m+1}(b{)}_{m+1}}{(m+1)!}{z}^{m+1}{}_2{F}_1(a+m+1,b+m+1;m+2;z)}$$

₂F₁(z) is the most common type ofgeneralized hypergeometric series_pF_q.

Using the identity${\displaystyle (a{)}_{n+1}=a(a+1{)}_n}$, it is shown that

$${\displaystyle \frac{d}{dz}{}_2{F}_1(a,b;c;z)=\frac{ab}{c}{}_2{F}_1(a+1,b+1;c+1;z)}$$

and more generally,

$${\displaystyle \frac{d^n}{d{z}^n}{}_2{F}_1(a,b;c;z)=\frac{(a{)}_n(b{)}_n}{(c{)}_n}{}_2{F}_1(a+n,b+n;c+n;z)}$$

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

Whena = 1 andb =c, the series reduces into a plaingeometric series, i.e.

hence, the namehypergeometric. This function can be considered as a generalization of thegeometric series.

Theconfluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function

$${\displaystyle M(a,c,z)=\underset{b\to \mathrm{\infty }}{lim}{}_2{F}_1(a,b;c;b^{-1}z)}$$

so all functions that are essentially special cases of it, such asBessel 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

$${\displaystyle {}_2{F}_1(a,1-a;c;z)=\mathrm{\Gamma }(c)z^{\frac{1-c}{2}}(1-z{)}^{\frac{c-1}{2}}{P}_{-a}^{1-c}(1-2z)}$$

Several orthogonal polynomials, includingJacobi polynomialsP^(α,β)
_n and their special casesLegendre polynomials,Chebyshev polynomials,Gegenbauer polynomials,Zernike polynomials can be written in terms of hypergeometric functions using

$${\displaystyle {}_2{F}_1(-n,\alpha +1+\beta +n;\alpha +1;x)=\frac{n!}{(\alpha +1{)}_n}{P}_n^{(\alpha ,\beta )}(1-2x)}$$

Other polynomials that are special cases includeKrawtchouk polynomials,Meixner polynomials,Meixner–Pollaczek polynomials.

Given${\displaystyle z\in \mathbb{C}\setminus \{0,1\}}$, let

$${\displaystyle \tau =\mathrm{i}\frac{{}_2{F}_1(\frac{1}{2},\frac{1}{2};1;1-z)}{{}_2{F}_1(\frac{1}{2},\frac{1}{2};1;z)}.}$$

Then

$${\displaystyle \lambda (\tau )=\frac{{\theta }_2(\tau {)}^4}{{\theta }_3(\tau {)}^4}=z}$$

is themodular lambda function, where

$${\displaystyle {\theta }_2(\tau )=\sum _{n\in \mathbb{Z}}{e}^{\pi i\tau (n+1/2{)}^2},{\theta }_3(\tau )=\sum _{n\in \mathbb{Z}}{e}^{\pi i\tau n^2}.}$$

Thej-invariant, amodular function, is a rational function in${\displaystyle \lambda (\tau )}$.

Incomplete beta functionsB_x(p,q) are related by

$${\displaystyle B_x(p,q)=\frac{x^p}{p}{}_2{F}_1(p,1-q;p+1;x).}$$

Thecomplete elliptic integralsK andE are given by^[1]

The hypergeometric function is a solution of Euler's hypergeometric differential equation

$${\displaystyle z(1-z)\frac{d^{2}w}{d{z}^2}+\left[c-(a+b+1)z\right]\frac{dw}{dz}-abw=0.}$$

which has threeregular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given byRiemann'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 to the hypergeometric differential equation are built out of the hypergeometric series₂F₁(a,b;c;z). The equation has twolinearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the formx^s times a holomorphic function ofx, wheres is one of the two roots of the indicial equation andx is a local variable vanishing at a regular singular point. This gives 3 × 2 = 6 special solutions, as follows.

Around the pointz = 0, two independent solutions are, ifc is not a non-positive integer,

$${\displaystyle {}_2{F}_1(a,b;c;z)}$$

and, on condition thatc is not an integer,

$${\displaystyle z^{1-c}{}_2{F}_1(1+a-c,1+b-c;2-c;z)}$$

Ifc is a non-positive integer 1 −m, then the first of these solutions does not exist and must be replaced by${\displaystyle z^{m}F(a+m,b+m;1+m;z).}$ The second solution does not exist whenc is an integer greater than 1, and is equal to the first solution, or its replacement, whenc is any other integer. So whenc 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 ofz, involving thedigamma function. SeeOlde Daalhuis (2010) for details.

Aroundz = 1, ifc − a − b is not an integer, one has two independent solutions

$${\displaystyle {}_2{F}_1(a,b;1+a+b-c;1-z)}$$

and

$${\displaystyle (1-z{)}^{c-a-b}{}_2{F}_1(c-a,c-b;1+c-a-b;1-z)}$$

Aroundz = ∞, ifa − b is not an integer, one has two independent solutions

$${\displaystyle z^{-a}{}_2{F}_1\left(a,1+a-c;1+a-b;z^{-1}\right)}$$

and

$${\displaystyle z^{-b}{}_2{F}_1\left(b,1+b-c;1+b-a;z^{-1}\right).}$$

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 calledconnection formulas.

A second orderFuchsian equation withn singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to theCoxeter group W(D_n) of order 2ⁿ⁻¹n!. The hypergeometric equation is the casen = 3, with group of order 24 isomorphic to the symmetric group on 4 points, as first described byKummer. 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 aKlein 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 solutionF(a,b;c;z) to one of

$${\displaystyle \begin{array}{r}(1-z{)}^{-a}F\left(a,c-b;c;\frac{z}{z-1}\right)\\ F(a,b;1+a+b-c;1-z)\\ (1-z{)}^{-b}F\left(c-a,b;c;\frac{z}{z-1}\right)\end{array}}$$

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 toF(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

The hypergeometric differential equation may be brought into the Q-form

$${\displaystyle \frac{d^{2}u}{d{z}^2}+Q(z)u(z)=0}$$

by making the substitutionu =wv and eliminating the first-derivative term. One finds that

$${\displaystyle Q=\frac{z^2[1-(a-b{)}^2]+z[2c(a+b-1)-4ab]+c(2-c)}{4z^2(1-z{)}^2}}$$

andv is given by the solution to

$${\displaystyle \frac{d}{dz}\log v(z)=-\frac{c-z(a+b+1)}{2z(1-z)}=-\frac{c}{2z}-\frac{1+a+b-c}{2(z-1)}}$$

which is

$${\displaystyle v(z)=z^{-c/2}(1-z{)}^{(c-a-b-1)/2}.}$$

The Q-form is significant in its relation to theSchwarzian derivative (Hille 1976, pp. 307–401).

TheSchwarz triangle maps orSchwarzs-functions are ratios of pairs of solutions.

$${\displaystyle s_k(z)=\frac{{\varphi }_k^{(1)}(z)}{{\varphi }_k^{(0)}(z)}}$$

wherek is one of the points 0, 1, ∞. The notation

$${\displaystyle D_k(\lambda ,\mu ,\nu ;z)=s_k(z)}$$

is also sometimes used. Note that the connection coefficients becomeMöbius transformations on the triangle maps.

Note that each triangle map isregular atz ∈ {0, 1, ∞} respectively, with

and$${\displaystyle s_{\mathrm{\infty }}(z)=z^{\nu }(1+\mathcal{O}(\frac{1}{z})).}$$

In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps areconformal maps of theupper half-planeH to triangles on theRiemann sphere, bounded by circular arcs. This mapping isa generalization of theSchwarz–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 integersp,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 ofautomorphic functions for thetriangle group 〈p, q, r〉 = Δ(p, q, r).

The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in thez plane that return to the same point.That is, when the path winds around a singularity of₂F₁, 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):

$${\displaystyle {\pi }_1(\mathbf{C}\setminus \{0,1\},z_0)\to \text{GL}(2,\mathbf{C})}$$

where π₁ is thefundamental group. In other words, the monodromy is a two dimensional linear representation of the fundamental group. Themonodromy 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.^[2] If (α, α'), (β, β') and (γ,γ') are the exponents at 0, 1 and ∞, then, takingz₀ near 0, the loops around 0 and 1 have monodromy matrices

where

$${\displaystyle \mu =\frac{\sin \pi (\alpha +{\beta }^{\mathrm{\prime }}+{\gamma }^{\mathrm{\prime }})\sin \pi ({\alpha }^{\mathrm{\prime }}+\beta +{\gamma }^{\mathrm{\prime }})}{\sin \pi ({\alpha }^{\mathrm{\prime }}+{\beta }^{\mathrm{\prime }}+{\gamma }^{\mathrm{\prime }})\sin \pi (\alpha +\beta +{\gamma }^{\mathrm{\prime }})}.}$$

If 1 −a,c −a −b,a −b are non-integerrational numbers with denominatorsk,l,m then the monodromy group is finiteif and only if${\displaystyle 1/k+1/l+1/m>1}$, seeSchwarz's list orKovacic's algorithm.

IfB is thebeta function then

$${\displaystyle \mathrm{B}(b,c-b){}_2{F}_1(a,b;c;z)={\int }_0^1{x}^{b-1}(1-x{)}^{c-b-1}(1-zx{)}^{-a}dx\mathrm{\Re }(c)>\mathrm{\Re }(b)>0,}$$

provided thatz 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 thebinomial theorem and then integrating term by term forz with absolute value smaller than 1, and by analytic continuation elsewhere. Whenz 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 impliesEuler's and Pfaff's hypergeometric transformations.

Other representations, corresponding to otherbranches, are given by taking the same integrand, but taking the path of integration to be a closedPochhammer cycle enclosing the singularities in various orders. Such paths correspond to themonodromy action.

Barnes used the theory ofresidues to evaluate theBarnes integral

$${\displaystyle \frac{1}{2\pi i}{\int }_{-i\mathrm{\infty }}^{i\mathrm{\infty }}\frac{\mathrm{\Gamma }(a+s)\mathrm{\Gamma }(b+s)\mathrm{\Gamma }(-s)}{\mathrm{\Gamma }(c+s)}(-z{)}^{s}ds}$$

as

$${\displaystyle \frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(c)}{}_2{F}_1(a,b;c;z),}$$

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.

The Gauss hypergeometric function can be written as aJohn transform (Gelfand, Gindikin & Graev 2003, 2.1.2).

The six functions

$${\displaystyle {}_2{F}_1(a\pm 1,b;c;z),{}_2{F}_1(a,b\pm 1;c;z),{}_2{F}_1(a,b;c\pm 1;z)}$$

are called contiguous to₂F₁(a,b;c;z). Gauss showed that₂F₁(a,b;c;z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms ofa,b,c, andz. This gives

$${\displaystyle \left(\begin{array}{c}6\\ 2\end{array}\right)=15}$$

relations, given by identifying any two lines on the right hand side of

whereF =₂F₁(a,b;c;z),F(a+) =₂F₁(a + 1,b;c;z), and so on. Repeatedly applying these relations gives a linear relation overC(z) between any three functions of the form

$${\displaystyle {}_2{F}_1(a+m,b+n;c+l;z),}$$

wherem,n, andl are integers.^[3][4]

Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:

$${\displaystyle \frac{{}_2{F}_1(a+1,b;c+1;z)}{{}_2{F}_1(a,b;c;z)}=\frac{1}{1+\frac{\frac{(a-c)b}{c(c+1)}z}{1+\frac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)}z}{1+\frac{\frac{(a-c-1)(b+1)}{(c+2)(c+3)}z}{1+\frac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)}z}{1+\ddots }}}}}}$$

Transformation formulas relate two hypergeometric functions at different values of the argumentz.

Euler's transformation is$${\displaystyle {}_2{F}_1(a,b;c;z)=(1-z{)}^{c-a-b}{}_2{F}_1(c-a,c-b;c;z).}$$It follows by combining the two Pfaff transformationswhich in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, seeRathie & Paris (2007) andRakha & Rathie (2011).It can also be written as linear combination

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 aquadratic transformation of the hypergeometric function, connecting it to a different value ofz related by a quadratic equation. The first examples were given byKummer (1836), and a complete list was given byGoursat (1881). A typical example is

$${\displaystyle {}_2{F}_1(a,b;2b;z)=(1-z{)}^{-\frac{a}{2}}{}_2{F}_1\left(\frac{1}{2}a,b-\frac{1}{2}a;b+\frac{1}{2};\frac{z^2}{4z-4}\right)}$$

If 1−c,a−b,a+b−c differ by signs or two of them are 1/3 or −1/3 then there is acubic transformation of the hypergeometric function, connecting it to a different value ofz related by a cubic equation. The first examples were given byGoursat (1881). A typical example is

$${\displaystyle {}_2{F}_1\left(\frac{3}{2}a,\frac{1}{2}(3a-1);a+\frac{1}{2};-\frac{z^2}{3}\right)=(1+z{)}^{1-3a}{}_2{F}_1\left(a-\frac{1}{3},a;2a;2z(3+z^2)(1+z{)}^{-3}\right)}$$

There are also some transformations of degree 4 and 6. Transformations of other degrees only exist ifa,b, andc are certain rational numbers (Vidunas 2005). For example,$${\displaystyle {}_2{F}_1\left(\frac{1}{4},\frac{3}{8};\frac{7}{8};z\right)(z^4-60z^3+134z^2-60z+1{)}^{1/16}={}_2{F}_1\left(\frac{1}{48},\frac{17}{48};\frac{7}{8};\frac{-432z(z-1{)}^2(z+1{)}^8}{(z^4-60z^3+134z^2-60z+1{)}^3}\right).}$$

SeeSlater (1966, Appendix III) for a list of summation formulas at special points, most of which also appear inBailey (1935).Gessel & Stanton (1982) gives further evaluations at more points.Koepf (1995) shows how most of these identities can be verified by computer algorithms.

Gauss's summation theorem, named forCarl Friedrich Gauss, is the identity

$${\displaystyle {}_2{F}_1(a,b;c;1)=\frac{\mathrm{\Gamma }(c)\mathrm{\Gamma }(c-a-b)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)},\mathrm{\Re }(c)>\mathrm{\Re }(a+b)}$$

which follows from Euler's integral formula by puttingz = 1. It includes theVandermonde identity as a special case.

For the special case where${\displaystyle a=-m}$,$${\displaystyle {}_2{F}_1(-m,b;c;1)=\frac{(c-b{)}_m}{(c{)}_m}}$$

Dougall's formula generalizes this to thebilateral hypergeometric series atz = 1.

There are many cases where hypergeometric functions can be evaluated atz = −1 by using a quadratic transformation to changez = −1 toz = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named forErnst Kummer:

$${\displaystyle {}_2{F}_1(a,b;1+a-b;-1)=\frac{\mathrm{\Gamma }(1+a-b)\mathrm{\Gamma }(1+\frac{1}{2}a)}{\mathrm{\Gamma }(1+a)\mathrm{\Gamma }(1+\frac{1}{2}a-b)}}$$

which follows from Kummer's quadratic transformations

and Gauss's theorem by puttingz = −1 in the first identity. For generalization of Kummer's summation, seeLavoie, Grondin & Rathie (1996).

Gauss's second summation theorem is

$${\displaystyle {}_2{F}_1\left(a,b;\frac{1}{2}\left(1+a+b\right);\frac{1}{2}\right)=\frac{\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}\left(1+a+b\right))}{\mathrm{\Gamma }(\frac{1}{2}\left(1+a)\right)\mathrm{\Gamma }(\frac{1}{2}\left(1+b\right))}.}$$

Bailey's theorem is

$${\displaystyle {}_2{F}_1\left(a,1-a;c;\frac{1}{2}\right)=\frac{\mathrm{\Gamma }(\frac{1}{2}c)\mathrm{\Gamma }(\frac{1}{2}\left(1+c\right))}{\mathrm{\Gamma }(\frac{1}{2}\left(c+a\right))\mathrm{\Gamma }(\frac{1}{2}\left(1+c-a\right))}.}$$

For generalizations of Gauss's second summation theorem and Bailey's summation theorem, seeLavoie, Grondin & Rathie (1996).

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 inGessel & Stanton (1982) andKoepf (1995). Some typical examples are given by

$${\displaystyle {}_2{F}_1\left(a,-a;\frac{1}{2};\frac{x^2}{4(x-1)}\right)=\frac{(1-x{)}^a+(1-x{)}^{-a}}{2},}$$

which can be restated as

$${\displaystyle T_a(\cos x)={}_2{F}_1\left(a,-a;\frac{1}{2};\frac{1}{2}(1-\cos x)\right)=\cos (ax)}$$

whenever −π <x <π andT is the (generalized)Chebyshev polynomial.

• Appell series
• Basic hypergeometric series
• Bilateral hypergeometric series
• Elliptic hypergeometric series
• General hypergeometric function
• Generalized hypergeometric series
• Hypergeometric distribution
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation

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