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{{Use American English|date = January 2019}}
{{Short description|Special function defined by a hypergeometric series}}
{{dablink| The term "hypergeometric function" sometimes refers to the [[generalized hypergeometric function]]. For other hypergeometric functions see [[#See also|See also]].}}

In [[mathematics]], the Gaussian or ordinary '''hypergeometric function''' <sub>2</sub>''F''<sub>1</sub>(''a'',''b'';''c'';''z'') is a [[Special functions|special function]] represented by the '''hypergeometric series''', that includes many other special functions as [[special case|specific]] or [[limiting case (mathematics)|limiting case]]s. It is a solution of a second-order [[linear function|linear]] [[ordinary differential equation]] (ODE). Every second-order linear ODE with three [[regular singular point]]s can be transformed into this equation.

For systematic lists of some of the many thousands of published [[Identity (mathematics)|identities]] involving the hypergeometric function, see the reference works by {{harvtxt | Erdélyi | Magnus | Oberhettinger | Tricomi |1953}} and {{harvtxt | 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 {{harvs|txt|authorlink=Carl Friedrich Gauss|first=Carl Friedrich|last=Gauss|year=1813}}.

Studies in the nineteenth century included those of {{harvs|txt|authorlink=Ernst Kummer|first=Ernst|last=Kummer|year=1836}}, and the fundamental characterisation by {{harvs|txt|authorlink=Bernhard Riemann|first=Bernhard|last=Riemann|year=1857}} of the hypergeometric function by means of the differential equation it satisfies.

Riemann showed that the second-order differential equation for <sub>2</sub>''F''<sub>1</sub>(''z''), examined in the complex plane, could be characterised (on the [[Riemann sphere]]) by its three [[regular singularity|regular singularities]].

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

==The hypergeometric series==
The hypergeometric function is defined for {{math|{{!}}''z''{{!}} < 1}} by the [[power series]]

:<math>{}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}.</math>

It is undefined (or infinite) if ''c'' equals a non-positive integer. Here (''q'')<sub>''n''</sub> is the (rising) [[Pochhammer symbol]], which is defined by:

:<math>(q)_n = \begin{cases}  1  & n = 0 \\
  q(q+1) \cdots (q+n-1) & n > 0
 \end{cases}</math>

The series terminates if either ''a'' or ''b'' is a nonpositive integer, in which case the function reduces to a polynomial:

:<math>{}_2F_1(-m,b;c;z) = \sum_{n=0}^m (-1)^n \binom{m}{n} \frac{(b)_n}{(c)_n} z^n.</math>

For complex arguments ''z'' with |''z''|&nbsp;≥&nbsp;1 it can be [[analytic continuation|analytically continued]] along any path in the complex plane that avoids the branch points 1 and infinity.

As {{math|''c'' → −''m''}}, where {{mvar|m}} is a non-negative integer, {{math|<sub>2</sub>''F''<sub>1</sub>(''z'') → ∞}}, but if we divide by [[gamma function|{{math|Γ(''c'')}}]], we have a limit:

:<math>\lim_{c\to -m}\frac{{}_2F_1(a,b;c;z)}{\Gamma(c)}=\frac{(a)_{m+1}(b)_{m+1}}{(m+1)!}z^{m+1}{}_2F_1(a+m+1,b+m+1;m+2;z)</math>

{{math|<sub>2</sub>''F''<sub>1</sub>(''z'')}} is the most usual type of [[generalized hypergeometric series]] {{math|''<sub>p</sub>F<sub>q</sub>''}}, and is often designated simply {{math|''F''(''z'')}}.

== Differentiation formulas ==
Using the identity <math> (a)_{n+1}=a (a+1)_n</math>, it is easily shown that

: <math>
\frac{d }{dz} \ {}_2F_1(a,b;c;z) = \frac{ab}{c} \ {}_2F_1(a+1,b+1;c+1;z)
</math>

and more generally,

: <math>
\frac{d^n }{dz^n} \ {}_2F_1(a,b;c;z) = \frac{(a)_n (b)_n}{(c)_n} \ {}_2F_1(a+n,b+n;c+n;z)
</math>

In the special case that <math>c = a + 1</math>, we have

: <math>
\frac{d }{dz} \ {}_2F_1(a,b;a+1;z) = \frac{d }{dz} \ {}_2F_1(b,a;a+1;z) = \frac{a((1-z)^{-b} - {}_2F_1(a,b;1+a;z))}{z}
</math>

==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

:<math>\ln(1+z) = z\ _2F_1(1,1;2;-z) </math>
:<math> (1-z)^{-a} = \ _2F_1(a,1;1;z) </math>
:<math>\arcsin(z) = z \ _2F_1\bigl(\tfrac{1}{2}, \tfrac{1}{2}; \tfrac{3}{2};z^2\bigr) </math>

The [[confluent hypergeometric function]] (or Kummer's function) can be given as a limit of the hypergeometric function

:<math>M(a,c,z) = \lim_{b\to \infty}{}_2F_1(a,b;c;b^{-1}z)</math>

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 function]]s 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

:<math>{}_2F_1(a,1-a;c;z) = \Gamma(c)z^{\tfrac{1-c}{2}}(1-z)^{\tfrac{c-1}{2}}P_{-a}^{1-c}(1-2z)</math>

Several orthogonal polynomials, including [[Jacobi polynomials]] ''P''{{su|p=(α,β)|b=''n''}} and their special cases [[Legendre polynomials]], [[Chebyshev polynomials]], [[Gegenbauer polynomials]] can be written in terms of hypergeometric functions using

:<math>{}_2F_1(-n,\alpha+1+\beta+n;\alpha+1;x) = \frac{n!}{(\alpha+1)_n}P^{(\alpha,\beta)}_n(1-2x)</math>

Other polynomials that are special cases include [[Krawtchouk polynomials]], [[Meixner polynomials]], [[Meixner–Pollaczek polynomials]].

[[Elliptic modular function]]s can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments ''a'', ''b'', ''c'' are 1, 1/2, 1/3, ... or 0. For example, if

:<math> \tau = {\rm{i}}\frac{{}_2F_1 \bigl( \frac{1}{2},\frac{1}{2};1;1-z \bigr)}{{}_2F_1 \bigl(\frac{1}{2},\frac{1}{2};1;z \bigr)}</math>

then

:<math> z = \kappa^2(\tau) = \frac{\theta_2(\tau)^4}{\theta_3(\tau)^4}</math>

is an elliptic modular function of τ.

[[Incomplete beta function]]s ''B''<sub>''x''</sub>(''p'',''q'') are related by

:<math> B_x(p,q) = \tfrac{x^p}{p}{}_2F_1(p,1-q;p+1;x)</math>

The [[complete elliptic integral]]s ''K'' and ''E'' are given by

:<math>K(k) = \tfrac{\pi}{2}\, _2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right)</math>
:<math>E(k) = \tfrac{\pi}{2}\, _2F_1\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^2\right)</math>

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

:<math>z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0.</math>

which has three [[regular singular point]]s: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by [[Riemann's differential equation]]. Any second order 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 <sub>2</sub>''F''<sub>1</sub>(''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 ''x''<sup>''s''</sup> 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 the regular singular point. This gives 3&nbsp;×&nbsp;2&nbsp;=&nbsp;6 special solutions, as follows.

Around the point ''z''&nbsp;=&nbsp;0, two independent solutions are, if ''c'' is not a non-positive integer,

:<math> \, _2F_1(a,b;c;z)</math>

and, on condition that ''c'' is not an integer,

:<math> z^{1-c} \, _2F_1(1+a-c,1+b-c;2-c;z)</math>

If ''c'' is a non-positive integer 1−''m'', then the first of these solutions does not exist and must be replaced by <math>z^mF(a+m,b+m;1+m;z).</math> 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 {{harvtxt|Olde Daalhuis|2010}} for details.

Around ''z''&nbsp;=&nbsp;1, if ''c''&nbsp;−&nbsp;''a''&nbsp;−&nbsp;''b'' is not an integer, one has two independent solutions

:<math>\, _2F_1(a,b;1+a+b-c;1-z)</math>

and

:<math> (1-z)^{c-a-b} \;_2F_1(c-a,c-b;1+c-a-b;1-z)</math>

Around ''z''&nbsp;=&nbsp;∞, if ''a''&nbsp;−&nbsp;''b'' is not an integer, one has two independent solutions

:<math> z^{-a}\, _2F_1 \left (a,1+a-c;1+a-b; z^{-1} \right)</math>

and

:<math> z^{-b}\, _2F_1 \left (b,1+b-c;1+b-a; z^{-1} \right ).</math>

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 ({{su|p=6|b=3}}) =&nbsp;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]] ''D''<sub>''n''</sub> of order ''n''!2<sup>''n''−1</sup>. For the hypergeometric equation ''n''=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by
[[Ernst Kummer|Kummer]]. The isomorphism with 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

:<math>(1-z)^{-a} F \left (a,c-b;c; \tfrac{z}{z-1} \right )</math>
:<math>F(a,b;1+a+b-c;1-z)</math>
:<math>(1-z)^{-b} F \left(c-a,b;c; \tfrac{z}{z-1} \right )</math>

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

:<math>{}_2F_1(a,b;c;z)  = (1-z)^{c-a-b} \, {}_2F_1(c-a,c-b;c;z) \ \ \  \text{Euler transformation} </math>
:<math>{}_2F_1(a,b;c;z)  =(1-z)^{-a} \, {}_2F_1(a,c-b;c; \tfrac{z}{z-1})\ \ \  \text{Pfaff transformation} </math>
:<math>{}_2F_1(a,b;c;z)  =(1-z)^{-b} \, {}_2F_1(c-a,b;c; \tfrac{z}{z-1})\ \ \  \text{Pfaff transformation} </math>

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

:<math>\frac{d^2u}{dz^2}+Q(z)u(z) = 0</math>

by making the substitution ''w'' = ''uv'' and eliminating the first-derivative term. One finds that

:<math>Q=\frac{z^2[1-(a-b)^2] +z[2c(a+b-1)-4ab] +c(2-c)}{4z^2(1-z)^2}</math>

and ''v'' is given by the solution to

:<math>\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)}</math>

which is

:<math>v(z)=z^{-c/2}(1-z)^{(c-a-b-1)/2}.</math>

The Q-form is significant in its relation to the [[Schwarzian derivative]] {{harv|Hille|1976|pp=307–401}}.

===Schwarz triangle maps===
{{see also|Schwarz triangle function}}
The '''Schwarz triangle maps''' or '''Schwarz ''s''-functions''' are ratios of pairs of solutions.

:<math>s_k(z) = \frac{\phi_k^{(1)}(z)}{\phi_k^{(0)}(z)}</math>

where ''k'' is one of the points 0, 1, ∞. The notation

:<math>D_k(\lambda,\mu,\nu;z)=s_k(z)</math>

is also sometimes used. Note that the connection coefficients become [[Möbius transformation]]s on the triangle maps.

Note that each triangle map is [[Regular singular point|regular]] at ''z'' ∈ {0, 1, ∞} respectively, with

:<math>s_0(z)=z^\lambda (1+\mathcal{O}(z))</math>
:<math>s_1(z)=(1-z)^\mu (1+\mathcal{O}(1-z))</math>
and
:<math>s_\infty(z)=z^\nu (1+\mathcal{O}(\tfrac{1}{z})).</math>

In the special case of λ, μ and ν real, with 0&nbsp;&le;&nbsp;λ,μ,ν&nbsp;&lt;&nbsp;1 then the s-maps are [[conformal map]]s of the [[upper half-plane]] '''H''' to triangles on the [[Riemann sphere]], bounded by circular arcs. This mapping is [[Schwarzian derivative#Conformal mapping of circular arc polygons|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 function]]s for the [[triangle group]] 〈''p'',&nbsp;''q'',&nbsp;''r''〉&nbsp;=&nbsp;Δ(''p'',&nbsp;''q'',&nbsp;''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  <sub>2</sub>''F''<sub>1</sub>, 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):

:<math>\pi_1(\mathbf{C}\setminus\{0,1\},z_0) \to \text{GL}(2,\mathbf{C})</math>

where π<sub>1</sub> 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.<ref>{{harvnb|Ince|1944|pages=393–393}}</ref> If (α, α'), (β, β') and  (γ,γ') are the exponents at 0, 1 and ∞, then, taking ''z''<sub>0</sub> near 0, the loops around 0 and 1 have monodromy matrices

:<math>g_0=\begin{pmatrix} e^{2\pi i\alpha} & 0\\ 0 & e^{2\pi i\alpha^\prime}\end{pmatrix}\,\,\,</math> and <math>\,\,\,g_1=\begin{pmatrix} {\mu e^{2\pi i \beta} 
-e^{2\pi i\beta^\prime}\over \mu -1} & {\mu (e^{2\pi i \beta} 
-e^{2\pi i\beta^\prime)}\over (\mu -1)^2}\\e^{2\pi i\beta^\prime} - e^{2\pi i\beta} & {\mu e^{2\pi i \beta^\prime} 
-e^{2\pi i\beta}\over \mu -1}\end{pmatrix},</math>

where

:<math>\mu = {\sin \pi(\alpha +\beta^\prime +\gamma^\prime) \sin \pi(\alpha^\prime + \beta+\gamma^\prime)\over \sin \pi(\alpha^\prime +  \beta^\prime +\gamma^\prime) \sin \pi(\alpha + \beta +\gamma^\prime)}.</math>

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 <math>1/k + 1/l + 1/m > 1</math>, see [[Schwarz's list]] or [[Picard–Vessiot theory|Kovacic's algorithm]].

==Integral formulas==

===Euler type===
If ''B'' is the [[beta function]] then

:<math>\Beta(b,c-b)\,_2F_1(a,b;c;z) = \int_0^1 x^{b-1} (1-x)^{c-b-1}(1-zx)^{-a} \, dx \qquad \real(c) > \real(b) > 0, </math>

provided that ''z'' is not a real number such that it is greater than or equal to 1. and can be proved by expanding (1&nbsp;−&nbsp;''zx'')<sup>−''a''</sup> 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&nbsp;−&nbsp;''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 [[principal branch|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 [[Residue (complex analysis)|residues]]  to evaluate the [[Barnes integral]]

:<math>\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)} (-z)^s \, ds</math>

as

:<math>\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,_2F_1(a,b;c;z),</math>

where the contour is drawn to separate the poles 0, 1, 2... from the poles −''a'', −''a''&nbsp;−&nbsp;1,&nbsp;..., −''b'', −''b''&nbsp;−&nbsp;1,&nbsp;...&nbsp;. 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]] {{harv|Gelfand|Gindikin|Graev|2003|loc=2.1.2}}.

==Gauss' contiguous relations==
The six functions

:<math>{}_2F_1 (a\pm 1,b;c;z), \quad  {}_2F_1 (a,b\pm 1;c;z), \quad {}_2F_1 (a,b;c\pm 1;z)</math>

are called contiguous to {{math|<sub>2</sub>''F''<sub>1</sub>(''a'', ''b''; ''c''; ''z'')}}. Gauss showed that {{math|<sub>2</sub>''F''<sub>1</sub>(''a'', ''b''; ''c''; ''z'')}} can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of {{math|''a'', ''b'', ''c''}}, and {{mvar|z}}. This gives

:<math> \begin{pmatrix} 6 \\ 2 \end{pmatrix} = 15</math>

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

:<math>\begin{align}
z\frac{dF}{dz} &= z\frac{ab}{c}F(a+,b+,c+) \\
&=a(F(a+)-F) \\
&=b(F(b+)-F) \\
&=(c-1)(F(c-)-F) \\
&=\frac{(c-a)F(a-)+(a-c+bz)F}{1-z} \\
&=\frac{(c-b)F(b-)+(b-c+az)F}{1-z} \\
&=z\frac{(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)}
\end{align}</math>

where {{math|''F'' {{=}} <sub>2</sub>''F''<sub>1</sub>(''a'', ''b''; ''c''; ''z''), ''F''(''a''+) {{=}} <sub>2</sub>''F''<sub>1</sub>(''a'' + 1, ''b''; ''c''; ''z'')}}, and so on. Repeatedly applying these relations gives a linear relation over {{math|'''C'''(z)}} between any three functions of the form

:<math>{}_2F_1 (a+m,b+n;c+l;z),</math>

where ''m'', ''n'', and ''l'' are integers.

===Gauss' continued fraction===
{{main|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:

:<math>\frac{{}_2F_1(a+1,b;c+1;z)}{{}_2F_1(a,b;c;z)} = \cfrac{1}{1 + \cfrac{\frac{(a-c)b}{c(c+1)} z}{1 + \cfrac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)} z}{1 + \cfrac{\frac{(a-c-1)(b+1)}{(c+2)(c+3)} z}{1 + \cfrac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)} z}{1 + {}\ddots}}}}}</math>

==Transformation formulas==
Transformation formulas relate two hypergeometric functions at different values of the argument ''z''.

===Fractional linear transformations===
Euler's transformation is
:<math>{}_2F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2F_1 (c-a, c-b;c ; z).</math>
It follows by combining the two Pfaff transformations
:<math>{}_2F_1 (a,b;c;z) = (1-z)^{-b} {}_2F_1 \left (b,c-a;c;\tfrac{z}{z-1} \right )</math>
:<math>{}_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 \left (a, c-b;c ; \tfrac{z}{z-1} \right )</math>
which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see {{harvtxt|Rathie|Paris|2007}} and {{harvtxt|Rakha|Rathie|2011}}.
It can also be written as linear combination
:<math>
\begin{align}
{}_2F_1(a,b;c,z) = {} & \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_2F_1(a,b;a+b+1-c;1-z) \\[6pt]
& {} + \frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b} {}_2F_1(c-a,c-b;1+c-a-b;1-z).
\end{align}
</math>

===Quadratic  transformations===
If two of the numbers 1&nbsp;−&nbsp;''c'', ''c''&nbsp;−&nbsp;1, ''a''&nbsp;−&nbsp;''b'', ''b''&nbsp;−&nbsp;''a'', ''a''&nbsp;+&nbsp;''b''&nbsp;−&nbsp;''c'', ''c''&nbsp;−&nbsp;''a''&nbsp;−&nbsp;''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 {{harvtxt|Kummer|1836}}, and a complete list was given by {{harvtxt|Goursat|1881}}. A typical example is

:<math>{}_2F_1(a,b;2b;z) = (1-z)^{-\frac{a}{2}} {}_2F_1 \left (\tfrac{1}{2}a, b-\tfrac{1}{2}a; b+\tfrac{1}{2}; \frac{z^2}{4z-4} \right)</math>

===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 {{harvtxt|Goursat|1881}}. A typical example is

:<math>{}_2F_1 \left (\tfrac{3}{2}a,\tfrac{1}{2}(3a-1);a+\tfrac{1}{2};-\tfrac{z^2}{3} \right) = (1+z)^{1-3a} \, {}_2F_1 \left (a-\tfrac{1}{3}, a; 2a; 2z(3+z^2)(1+z)^{-3} \right )</math>

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 {{harv|Vidunas|2005}}.  For example,
:<math>{}_2F_1 \left (\tfrac{1}{4},\tfrac{3}{8};\tfrac{7}{8}; z \right) (z^4-60z^3+134z^2-60z+1)^{1/16}  =
  {}_2F_1 \left (\tfrac{1}{48}, \tfrac{17}{48}; \tfrac{7}{8}; \tfrac{-432 z (z-1)^2 (z+1)^8}{(z^4-60z^3+134z^2-60z+1)^3} \right ).</math>

==Values at special points ''z''==
See {{harvtxt|Slater|1966|loc=Appendix III}} for a list of summation formulas at special points, most of which also appear in {{harvtxt|Bailey|1935}}.  {{harvtxt | Gessel | Stanton | 1982}} gives further evaluations at more points. {{harvtxt|Koepf|1995}} shows how most of these identities can be verified by computer algorithms.

===Special values at ''z''&nbsp;=&nbsp;1===
Gauss's theorem, named for [[Carl Friedrich Gauss]], is the identity

:<math>{}_2F_1 (a,b;c;1)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}, \qquad   \Re(c)>\Re(a+b) </math>

which follows from Euler's integral formula by putting ''z''&nbsp;=&nbsp;1. It includes the [[Vandermonde identity]] as a special case.

For the special case where <math> a=-m </math>, 
:<math>{}_2F_1 (-m,b;c;1)=\frac{ (c-b)_{m} }{(c)_{m}  } </math>

[[bilateral hypergeometric series|Dougall's formula]] generalizes this to the [[bilateral hypergeometric series]] at ''z''&nbsp;=&nbsp;1.

===Kummer's theorem (''z''&nbsp;=&nbsp;−1) ===
<span id="Kummer's theorem"></span>There are many cases where hypergeometric functions can be evaluated at ''z''&nbsp;=&nbsp;−1 by using a quadratic transformation to change ''z''&nbsp;=&nbsp;−1 to ''z''&nbsp;=&nbsp;1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for [[Ernst Kummer]]:

:<math>{}_2F_1 (a,b;1+a-b;-1)= \frac{\Gamma(1+a-b)\Gamma(1+\tfrac12a)}{\Gamma(1+a)\Gamma(1+\tfrac12a-b)}</math>

which follows from Kummer's quadratic transformations

:<math>\begin{align}
_2F_1(a,b;1+a-b;z)&= (1-z)^{-a} \;_2F_1 \left(\frac a 2, \frac{1+a}2-b; 1+a-b; -\frac{4z}{(1-z)^2}\right)\\
&=(1+z)^{-a} \, _2F_1\left(\frac a 2, \frac{a+1}2; 1+a-b; \frac{4z}{(1+z)^2}\right)
\end{align}</math>

and Gauss's theorem by putting ''z''&nbsp;=&nbsp;−1 in the first identity. For generalization of Kummer's summation, see {{harvtxt | Lavoie | Grondin | Rathie | 1996}}.

===Values at ''z''&nbsp;=&nbsp;1/2===
Gauss's second summation theorem is

:<math>_2F_1 \left(a,b;\tfrac12\left(1+a+b\right);\tfrac12\right) = \frac{\Gamma(\tfrac12)\Gamma(\tfrac12\left(1+a+b\right))}{\Gamma(\tfrac12\left(1+a)\right)\Gamma(\tfrac12\left(1+b\right))}. </math>

Bailey's theorem is

:<math>_2F_1 \left(a,1-a;c;\tfrac12\right)= \frac{\Gamma(\tfrac12c)\Gamma(\tfrac12\left(1+c\right))}{\Gamma(\tfrac12\left(c+a\right))\Gamma(\tfrac12\left(1+c-a\right))}.</math>

For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see {{harvtxt | 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 {{harvtxt | Gessel | Stanton | 1982}} and {{harvtxt|Koepf|1995}}. Some typical examples are given by

:<math>{}_2F_1 \left(a,-a;\tfrac{1}{2};\tfrac{x^2}{4(x-1)} \right )  = \frac{(1-x)^a+(1-x)^{-a}}{2},</math>

which can be restated as

:<math>T_a(\cos x)={}_2F_1\left(a,-a;\tfrac{1}{2};\tfrac{1}{2}(1-\cos x)\right)=\cos(a x)</math>

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]] <sub>''p''</sub>H<sub>''p''</sub> are similar to generalized hypergeometric series, but summed over all integers
*[[Binomial series]] <sub>1</sub>F<sub>0</sub>
*[[Confluent hypergeometric series]] <sub>1</sub>F<sub>1</sub>(''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 <sub>2</sub>''F''<sub>1</sub>
*[[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 [[Israel Gelfand|I. M. Gelfand]]. 
*[[Generalized hypergeometric series]] <sub>''p''</sub>F<sub>''q''</sub> 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 <sub>''p''</sub>F<sub>''q''</sub> to the case ''p''>''q''+1.
*[[Meijer G-function]], an extension of the generalized hypergeometric series <sub>''p''</sub>F<sub>''q''</sub> 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 block]]s, special functions in [[two-dimensional conformal field theory]] that reduce to hypergeometric functions in some cases.

==References==
{{reflist}}
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* [[Frits Beukers|Beukers, Frits]] (2002), ''[http://www.math.uu.nl/people/beukers/MRIcourse93.ps Gauss' hypergeometric function]''. (lecture notes reviewing basics, as well as triangle maps and monodromy)
* {{dlmf | first= Adri B. | last= Olde Daalhuis | id= 15}}
* {{cite book | last1= Erdélyi | first1= Arthur | author1-link= Arthur Erdélyi | last2= Magnus | first2= Wilhelm | author2-link= Wilhelm Magnus | last3= Oberhettinger | first3= Fritz | lastauthoramp= yes | last4= Tricomi | first4= Francesco G. | title= Higher transcendental functions | volume= Vol. I | location= New York – Toronto – London | publisher= McGraw–Hill Book Company, Inc. | year= 1953 | isbn= 978-0-89874-206-0 | mr= 0058756 | url= http://apps.nrbook.com/bateman/Vol1.pdf | ref= harv }}
* Gasper, George & [[Mizan Rahman|Rahman, Mizan]] (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. {{ISBN|0-521-83357-4}}.
* {{cite journal | last= Gauss | first= Carl Friedrich | authorlink= Carl Friedrich Gauss | title= Disquisitiones generales circa seriem infinitam &nbsp; <math> 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.} </math> | language= Latin | url= https://books.google.com/books?id=uDMAAAAAQAAJ | location= Göttingen | journal= Commentationes societatis regiae scientarum Gottingensis recentiores | year= 1813 | volume= 2 | ref= harv}} 
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* {{cite journal | last1= Gessel | first1= Ira | lastauthoramp= yes | last2= Stanton | first2= Dennis | title= Strange evaluations of hypergeometric series | journal= SIAM Journal on Mathematical Analysis | year= 1982 | volume= 13 | issue= 2 | pages= 295–308 | issn= 0036-1410 | doi= 10.1137/0513021 | mr= 647127 | ref= harv}}
* {{cite journal |last=Goursat |first=Édouard |authorlink=Édouard Goursat |title=Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique |language=French |url=http://www.numdam.org/item?id=ASENS_1881_2_10__S3_0 |accessdate=2008-10-16 |journal=Annales Scientifiques de l'École Normale Supérieure |volume=10 |year=1881 |pages=3–142 |ref=harv }}
* {{cite book | last1= Heckman | first1= Gerrit | last2= Schlichtkrull | first2= Henrik | lastauthoramp= yes | title= Harmonic Analysis and Special Functions on Symmetric Spaces | location= San Diego | publisher= Academic Press | year= 1994 | isbn= 0-12-336170-2 | ref= harv}} (part 1 treats hypergeometric functions on Lie groups)
* {{cite book | last=Hille | first=Einar | year=1976 | title=Ordinary differential equations in the complex domain | url=https://archive.org/details/ordinarydifferen00hill_0 | url-access=registration | publisher=Dover | ISBN=0-486-69620-0 | ref=harv}}
*{{cite book|last=Ince|first=E. L.|authorlink=E. L. Ince|title=Ordinary Differential Equations|url=https://archive.org/details/in.ernet.dli.2015.476224|publisher= Dover Publications|year= 1944}}
* {{cite book | last= Klein | first= Felix | title= Vorlesungen über die hypergeometrische Funktion | language= German | url= http://resolver.sub.uni-goettingen.de/purl?PPN375394591 | location= Berlin, New York | publisher= Springer-Verlag | year= 1981 | series= Grundlehren der Mathematischen Wissenschaften | volume= 39 | isbn= 978-3-540-10455-1 | mr= 668700 | ref= harv}}
* {{cite journal | last= Koepf | first= Wolfram | title= Algorithms for m-fold hypergeometric summation | year= 1995 | journal= Journal of Symbolic Computation | issn= 0747-7171 | volume= 20 | issue= 4 | pages= 399–417 | mr=1384455 | doi= 10.1006/jsco.1995.1056 | ref= harv}}
* {{cite journal | last= Kummer | first= Ernst Eduard | title= Über die hypergeometrische Reihe&nbsp;<math> 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 +\mbox{etc.} </math> | language= German | url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN00214056X | journal= [[Journal für die reine und angewandte Mathematik]] | year= 1836 | volume= 15 | pages= 39–83, 127–172 | issn= 0075-4102 | ref= harv}}
*{{cite journal | last1=Lavoie | first1=J. L. | last2=Grondin | first2=F. | last3=Rathie | first3=A.K. | year=1996 | title= Generalizations of Whipple's theorem on the sum of a <sub>3</sub>F<sub>2</sub> | journal = J. Comput. Appl. Math. | volume=72 | pages=293–300 | ref=harv}}
* {{cite book | last1= Press | first1= W.H. | last2= Teukolsky | first2= S.A. | last3= Vetterling | first3= W.T. | last4= Flannery | first4= B.P. | lastauthoramp= yes | year= 2007 | title= Numerical Recipes: The Art of Scientific Computing | edition= 3rd | publisher= Cambridge University Press | publication-place= New York | isbn= 978-0-521-88068-8 | chapter= Section 6.13. Hypergeometric Functions | chapter-url= http://apps.nrbook.com/empanel/index.html#pg=318 | ref= harv}}
*{{cite journal | last1=Rakha | first1=M.A. | last2=Rathie | first2=Arjun K. | year=2011 | title=Extensions of Euler's type-II transformation and Saalschutz's theorem | journal= Bull. Korean Math. Soc. | volume=48 | issue=1 | pages=151–156 | ref=harv}}
*{{cite journal | last1=Rathie | first1= Arjun K. | last2= Paris | first2=R.B. | year=2007 |title= An extension of the Euler's-type transformation for the 3F2 series | journal= Far East J. Math. Sci. | volume=27 | issue=1 | pages=43–48 | ref=harv}}
* {{cite journal | last= Riemann | first= Bernhard | author-link= Bernhard Riemann | year= 1857 | title= Beiträge zur Theorie der durch die Gauss'sche Reihe ''F(α, β, γ, x)'' darstellbaren Functionen | journal= Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen | language= German | volume= 7 | pages= 3–22 | publisher= Verlag der Dieterichschen Buchhandlung | location= Göttingen | url= http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002018691 | ref= harv}} (a reprint of this paper can be found in {{cite web|url= http://www.emis.de/classics/Riemann/PFunct.pdf |title=All publications of Riemann }})
* {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Confluent hypergeometric functions | url= https://archive.org/details/confluenthyperge0000slat | url-access= registration | location= Cambridge, UK | publisher= Cambridge University Press | year= 1960 | mr= 0107026 | ref= harv}}
* {{cite book | last= Slater | first= Lucy Joan | title= Generalized hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | ref= harv}} (there is a 2008 paperback with {{ISBN|978-0-521-09061-2}})
* {{cite journal | last= Vidunas | first = Raimundas | title = Transformations of some Gauss hypergeometric functions | year = 2005 | journal = Journal of Symbolic Computation | volume = 178 | pages = 473–487 | doi=10.1016/j.cam.2004.09.053 | ref=harv| arxiv = math/0310436 }}
* {{cite book | last= Wall | first= H.S. | title= Analytic Theory of Continued Fractions | publisher= D. Van Nostrand Company, Inc. | year= 1948 | ref= harv}}
* {{cite book | last1= Whittaker | first1= E.T. | last2= Watson | first2= G.N. | lastauthoramp= yes | title= A Course of Modern Analysis | location= Cambridge, UK | publisher= Cambridge University Press | year= 1927 | ref= harv}}
* {{cite book | last= Yoshida | first= Masaaki | title= Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces | location= Braunschweig – Wiesbaden | publisher= Friedr. Vieweg & Sohn | year= 1997 | isbn= 3-528-06925-2 | mr= 1453580 | ref= harv}}

==External links==
* {{springer|title=Hypergeometric function|id=p/h048450}}
* John Pearson, [http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf Computation of Hypergeometric Functions] ([[University of Oxford]], MSc Thesis)
* Marko Petkovsek, Herbert Wilf and Doron Zeilberger, [https://web.archive.org/web/20060129095451/http://www.cis.upenn.edu/~wilf/AeqB.html The book "A = B"] (freely downloadable)
* {{MathWorld |title=Hypergeometric Function |urlname= HypergeometricFunction}}

{{Series (mathematics)}}

{{DEFAULTSORT:Hypergeometric Function}}
[[Category:Factorial and binomial topics]]
[[Category:Hypergeometric functions|*]]
[[Category:Ordinary differential equations]]
[[Category:Mathematical series]]