hypergeometric function

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 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 function is defined for {{math|{{!}}z{{!}} < 1}} by the power series

$$\{\}\_2F\_1(a,b;c;z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(a)\_n\; (b)\_n\}\{(c)\_n\}\; \backslash frac\{z^n\}\{n!\}\; =\; 1\; +\; \backslash frac\{ab\}\{c\}\backslash frac\{z\}\{1!\}\; +\; \backslash frac\{a(a+1)b(b+1)\}\{c(c+1)\}\backslash frac\{z^2\}\{2!\}\; +\; \backslash cdots.$$

It is undefined (or infinite) if {{mvar|c}} equals a non-positive integer. Here {{math|(q)_n}} is the (rising) Pochhammer symbol,{{NoteTag|This convention is common in hypergeometric function theory, but it is the opposite convention to the one used in Falling and rising factorials.}} which is defined by:

q(q+1) \cdots (q+n-1) & n > 0

\end{cases}

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

$$\{\}\_2F\_1(-m,b;c;z)\; =\; \backslash sum\_\{n=0\}^m\; (-1)^n\; \backslash binom\{m\}\{n\}\; \backslash frac\{(b)\_n\}\{(c)\_n\}\; z^n.$$

For complex arguments {{mvar|z}} with {{math|{{abs|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 {{math|z ≥ 1}}.

As {{math|c → −m}}, where {{mvar|m}} is a non-negative integer, one has {{math|₂F₁(z) → ∞}}. Dividing by the value {{math|Γ(c)}} of the gamma function, we have the limit:

$$\backslash lim\_\{c\backslash to\; -m\}\backslash frac\{\{\}\_2F\_1(a,b;c;z)\}\{\backslash Gamma(c)\}=\backslash frac\{(a)\_\{m+1\}(b)\_\{m+1\}\}\{(m+1)!\}z^\{m+1\}\{\}\_2F\_1(a+m+1,b+m+1;m+2;z)$$

{{math|₂F₁(z)}} is the most common type of generalized hypergeometric series {{mvar|_pF_q}}, and is often designated simply {{math|F(z)}}.

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

\frac{d }{dz} \ {}_2F_1(a,b;c;z) = \frac{a b}{c} \ {}_2F_1(a+1,b+1;c+1;z)

and more generally,

\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)

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

$$\backslash begin\{align\}$$

_2F_1\left(1, 1; 2; -z\right) &= \frac{\ln(1+z)}{z} \\

_2F_1(a, b; b; z) &= (1-z)^{-a} \quad (b \text{ arbitrary}) \\

_2F_1\left(\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; z^2\right) &= \frac{\arcsin(z)}{z} \\

\,_2F_1\left(\frac{1}{3}, \frac{2}{3}; \frac{3}{2}; -\frac{27x^2}{4}\right) &= \frac{\sqrt[3]{\frac{3x\sqrt{3}+\sqrt{27x^2+4}}{2}}-\sqrt[3]{\frac{2}{3x\sqrt{3}+\sqrt{27x^2+4}}}}{x\sqrt{3}} \\

\end{align}

When a=1 and b=c, the series reduces into a plain geometric series, i.e.

$$\backslash begin\{align\}$$

_2F_1\left(1, b; b; z\right) &= {}_1F_0\left(1;;z\right) = 1 + z + z^2 + z^3 + z^4 + \cdots \end{align}

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

$$M(a,c,z)\; =\; \backslash lim\_\{b\backslash to\backslash infty\}\{\}\_2F\_1(a,b;c;b^\{-1\}z)$$

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

$$\{\}\_2F\_1(a,1-a;c;z)\; =\; \backslash Gamma(c)z^\{\backslash tfrac\{1-c\}\{2\}\}(1-z)^\{\backslash tfrac\{c-1\}\{2\}\}P\_\{-a\}^\{1-c\}(1-2z)$$

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

$$\{\}\_2F\_1(-n,\backslash alpha+1+\backslash beta+n;\backslash alpha+1;x)\; =\; \backslash frac\{n!\}\{(\backslash alpha+1)\_n\}P^\{(\backslash alpha,\backslash beta)\}\_n(1-2x)$$

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

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

$$\backslash tau\; =\; \{\backslash rm\{i\}\}\backslash frac\{\{\}\_2F\_1\; \backslash bigl(\; \backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\};1;1-z\; \backslash bigr)\}\{\{\}\_2F\_1\; \backslash bigl(\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\};1;z\; \backslash bigr)\}.$$

Then

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

is the modular lambda function, where

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

The j-invariant, a modular function, is a rational function in $\backslash lambda\; (\backslash tau)$.

Incomplete beta functions B_x(p,q) are related by

$$B\_x(p,q)\; =\; \backslash tfrac\{x^p\}\{p\}\{\}\_2F\_1(p,1-q;p+1;x).$$

The complete elliptic integrals K and E are given by{{cite journal|first1=Tohru|last1=Morita|title=Use of the Gauss contiguous relations in computing the hypergeometric functions F(n+1/2,n+1/2;m;z)|year=1996|volume=2|number=1|pages=63–74|journal=Interd. Inf. Sci.|doi=10.4036/iis.1996.63|mr=1398101}}

$$\backslash begin\{align\}$$

K(k) &= \tfrac{\pi}{2}\, _2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right), \\

E(k) &= \tfrac{\pi}{2}\, _2F_1\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^2\right).

\end{align}

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

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

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 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 x^s 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,

$$\backslash ,\; \_2F\_1(a,b;c;z)$$

and, on condition that c is not an integer,

$$z^\{1-c\}\; \backslash ,\; \_2F\_1(1+a-c,1+b-c;2-c;z)$$

If c is a non-positive integer 1−m, then the first of these solutions does not exist and must be replaced by $z^mF(a+m,b+m;1+m;z).$ 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 = 1, if c − a − b is not an integer, one has two independent solutions

$$\backslash ,\; \_2F\_1(a,b;1+a+b-c;1-z)$$

and

$$(1-z)^\{c-a-b\}\; \backslash ;\_2F\_1(c-a,c-b;1+c-a-b;1-z)$$

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

$$z^\{-a\}\backslash ,\; \_2F\_1\; \backslash left\; (a,1+a-c;1+a-b;\; z^\{-1\}\; \backslash right)$$

and

$$z^\{-b\}\backslash ,\; \_2F\_1\; \backslash left\; (b,1+b-c;1+b-a;\; z^\{-1\}\; \backslash right\; ).$$

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 ({{su|p=6|b=3}}) = 20 linear relations between them called connection formulas.

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(D_n) 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

$$\backslash begin\{align\}$$

(1-z)^{-a} F \left (a,c-b;c; \tfrac{z}{z-1} \right ) \\

F(a,b;1+a+b-c;1-z) \\

(1-z)^{-b} F \left(c-a,b;c; \tfrac{z}{z-1} \right )

\end{align}

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

$$\backslash begin\{align\}$$

{}_2F_1(a,b;c;z) &= (1-z)^{c-a-b} \, {}_2F_1(c-a,c-b;c;z) && \text{Euler transformation} \\

{}_2F_1(a,b;c;z) &= (1-z)^{-a} \, {}_2F_1(a,c-b;c; \tfrac{z}{z-1}) && \text{Pfaff transformation} \\

{}_2F_1(a,b;c;z) &= (1-z)^{-b} \, {}_2F_1(c-a,b;c; \tfrac{z}{z-1}) && \text{Pfaff transformation}

\end{align}

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

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

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

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

and v is given by the solution to

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

which is

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

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

{{Main|Schwarz triangle function}}

The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.

$$s\_k(z)\; =\; \backslash frac\{\backslash phi\_k^\{(1)\}(z)\}\{\backslash phi\_k^\{(0)\}(z)\}$$

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

$$D\_k(\backslash lambda,\backslash mu,\backslash nu;z)=s\_k(z)$$

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

$$\backslash begin\{align\}$$

s_0(z) &= z^\lambda (1+\mathcal{O}(z)) \\

s_1(z) &= (1-z)^\mu (1+\mathcal{O}(1-z))

\end{align}

and

$$s\_\backslash infty(z)=z^\backslash nu\; (1+\backslash mathcal\{O\}(\backslash tfrac\{1\}\{z\})).$$

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).

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):

$$\backslash pi\_1(\backslash mathbf\{C\}\backslash setminus\backslash \{0,1\backslash \},z\_0)\; \backslash to\; \backslash text\{GL\}(2,\backslash mathbf\{C\})$$

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.{{harvnb|Ince|1944|pages=393–393}} 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

$$\backslash begin\{align\}$$

g_0 &= \begin{pmatrix} e^{2\pi i\alpha} & 0\\ 0 & e^{2\pi i\alpha^\prime}\end{pmatrix} \\

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},

\end{align}

where

$$\backslash mu\; =\; \{\backslash sin\; \backslash pi(\backslash alpha\; +\backslash beta^\backslash prime\; +\backslash gamma^\backslash prime)\; \backslash sin\; \backslash pi(\backslash alpha^\backslash prime\; +\; \backslash beta+\backslash gamma^\backslash prime)\backslash over\; \backslash sin\; \backslash pi(\backslash alpha^\backslash prime\; +\; \backslash beta^\backslash prime\; +\backslash gamma^\backslash prime)\; \backslash sin\; \backslash pi(\backslash alpha\; +\; \backslash beta\; +\backslash gamma^\backslash prime)\}.$$

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 $1/k\; +\; 1/l\; +\; 1/m\; >\; 1$, see Schwarz's list or Kovacic's algorithm.

If B is the beta function then

$$\backslash Beta(b,c-b)\backslash ,\_2F\_1(a,b;c;z)\; =\; \backslash int\_0^1\; x^\{b-1\}\; (1-x)^\{c-b-1\}(1-zx)^\{-a\}\; \backslash ,\; dx\; \backslash qquad\; \backslash real(c)\; >\; \backslash real(b)\; >\; 0,$$

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 used the theory of residues to evaluate the Barnes integral

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

as

$$\backslash frac\{\backslash Gamma(a)\backslash Gamma(b)\}\{\backslash Gamma(c)\}\backslash ,\_2F\_1(a,b;c;z),$$

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 a John transform {{harv|Gelfand|Gindikin|Graev|2003|loc=2.1.2}}.

The six functions

$$\{\}\_2F\_1\; (a\backslash pm\; 1,b;c;z),\; \backslash quad\; \{\}\_2F\_1\; (a,b\backslash pm\; 1;c;z),\; \backslash quad\; \{\}\_2F\_1\; (a,b;c\backslash pm\; 1;z)$$

are called contiguous to {{math|₂F₁(a, b; c; z)}}. Gauss showed that {{math|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 {{math|a, b, c}}, and {{mvar|z}}. This gives

$$\backslash begin\{pmatrix\}\; 6\; \backslash \backslash \; 2\; \backslash end\{pmatrix\}\; =\; 15$$

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

$$\backslash begin\{align\}$$

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}

where {{math|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 {{math|C(z)}} between any three functions of the form

$$\{\}\_2F\_1\; (a+m,b+n;c+l;z),$$

where m, n, and l are integers.{{cite journal|first1=Medhat A. |last1=Rakha |first2=Arjun K.|last2=Rathie|first3= Purnima |last3=Chopra|title=On some new contiguous relations for the Gauss hypergeometric function with applications|year=2011|journal=Comput. Math. Appl.|volume=61|issue=3 |pages=620–629|doi=10.1016/j.camwa.2010.12.008|mr=2764057}}

{{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:

$$\backslash frac\{\{\}\_2F\_1(a+1,b;c+1;z)\}\{\{\}\_2F\_1(a,b;c;z)\}\; =\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{\backslash frac\{(a-c)b\}\{c(c+1)\}\; z\}\{1\; +\; \backslash cfrac\{\backslash frac\{(b-c-1)(a+1)\}\{(c+1)(c+2)\}\; z\}\{1\; +\; \backslash cfrac\{\backslash frac\{(a-c-1)(b+1)\}\{(c+2)(c+3)\}\; z\}\{1\; +\; \backslash cfrac\{\backslash frac\{(b-c-2)(a+2)\}\{(c+3)(c+4)\}\; z\}\{1\; +\; \{\}\backslash ddots\}\}\}\}\}$$

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

Euler's transformation is

$$\{\}\_2F\_1\; (a,b;c;z)\; =\; (1-z)^\{c-a-b\}\; \{\}\_2F\_1\; (c-a,\; c-b;c\; ;\; z).$$

It follows by combining the two Pfaff transformations

$$\backslash begin\{align\}$$

{}_2F_1 (a,b;c;z) &= (1-z)^{-b} {}_2F_1 \left (b,c-a;c;\tfrac{z}{z-1} \right ) \\

{}_2F_1 (a,b;c;z) &= (1-z)^{-a} {}_2F_1 \left (a, c-b;c ; \tfrac{z}{z-1} \right ) \\

\end{align}

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

\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}

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 {{harvtxt|Kummer|1836}}, and a complete list was given by {{harvtxt|Goursat|1881}}. A typical example is

$$\{\}\_2F\_1(a,b;2b;z)\; =\; (1-z)^\{-\backslash frac\{a\}\{2\}\}\; \{\}\_2F\_1\; \backslash left\; (\backslash tfrac\{1\}\{2\}a,\; b-\backslash tfrac\{1\}\{2\}a;\; b+\backslash tfrac\{1\}\{2\};\; \backslash frac\{z^2\}\{4z-4\}\; \backslash right)$$

If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is 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

$$\{\}\_2F\_1\; \backslash left\; (\backslash tfrac\{3\}\{2\}a,\backslash tfrac\{1\}\{2\}(3a-1);a+\backslash tfrac\{1\}\{2\};-\backslash tfrac\{z^2\}\{3\}\; \backslash right)\; =\; (1+z)^\{1-3a\}\; \backslash ,\; \{\}\_2F\_1\; \backslash left\; (a-\backslash tfrac\{1\}\{3\},\; a;\; 2a;\; 2z(3+z^2)(1+z)^\{-3\}\; \backslash right\; )$$

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,

$$\{\}\_2F\_1\; \backslash left\; (\backslash tfrac\{1\}\{4\},\backslash tfrac\{3\}\{8\};\backslash tfrac\{7\}\{8\};\; z\; \backslash right)\; (z^4-60z^3+134z^2-60z+1)^\{1/16\}\; =$$

{}_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 ).

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.

Gauss's summation theorem, named for Carl Friedrich Gauss, is the identity

$$\{\}\_2F\_1\; (a,b;c;1)=\; \backslash frac\{\backslash Gamma(c)\backslash Gamma(c-a-b)\}\{\backslash Gamma(c-a)\backslash Gamma(c-b)\},\; \backslash qquad\; \backslash Re(c)>\backslash Re(a+b)$$

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 $a=-m$,

$$\{\}\_2F\_1\; (-m,b;c;1)=\backslash frac\{\; (c-b)\_\{m\}\; \}\{(c)\_\{m\}\; \}$$

Dougall's formula generalizes this to the bilateral hypergeometric series at 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:

$$\{\}\_2F\_1\; (a,b;1+a-b;-1)=\; \backslash frac\{\backslash Gamma(1+a-b)\backslash Gamma(1+\backslash tfrac12a)\}\{\backslash Gamma(1+a)\backslash Gamma(1+\backslash tfrac12a-b)\}$$

which follows from Kummer's quadratic transformations

$$\backslash begin\{align\}$$

_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}

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

Gauss's second summation theorem is

$$\_2F\_1\; \backslash left(a,b;\backslash tfrac12\backslash left(1+a+b\backslash right);\backslash tfrac12\backslash right)\; =\; \backslash frac\{\backslash Gamma(\backslash tfrac12)\backslash Gamma(\backslash tfrac12\backslash left(1+a+b\backslash right))\}\{\backslash Gamma(\backslash tfrac12\backslash left(1+a)\backslash right)\backslash Gamma(\backslash tfrac12\backslash left(1+b\backslash right))\}.$$

Bailey's theorem is

$$\_2F\_1\; \backslash left(a,1-a;c;\backslash tfrac12\backslash right)=\; \backslash frac\{\backslash Gamma(\backslash tfrac12c)\backslash Gamma(\backslash tfrac12\backslash left(1+c\backslash right))\}\{\backslash Gamma(\backslash tfrac12\backslash left(c+a\backslash right))\backslash Gamma(\backslash tfrac12\backslash left(1+c-a\backslash right))\}.$$

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

$$\{\}\_2F\_1\; \backslash left(a,-a;\backslash tfrac\{1\}\{2\};\backslash tfrac\{x^2\}\{4(x-1)\}\; \backslash right\; )\; =\; \backslash frac\{(1-x)^a+(1-x)^\{-a\}\}\{2\},$$

which can be restated as

$$T\_a(\backslash cos\; x)=\{\}\_2F\_1\backslash left(a,-a;\backslash tfrac\{1\}\{2\};\backslash tfrac\{1\}\{2\}(1-\backslash cos\; x)\backslash right)=\backslash cos(a\; x)$$

whenever −π < x < π and T 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

{{sfn whitelist|CITEREFOlde_Daalhuis2010}}

{{reflist}}

{{Notelist}}

• {{cite book | last1= Andrews | first1= George E. | author-link= George Andrews (mathematician) | last2= Askey | first2= Richard | last3= Roy | first3= Ranjan | name-list-style= amp | title= Special functions | publisher= Cambridge University Press | year= 1999 | series= Encyclopedia of Mathematics and its Applications | volume= 71 | isbn= 978-0-521-62321-6 | mr= 1688958 }}
• {{cite book | last=Bailey | first=W.N. | year=1935 | title=Generalized Hypergeometric Series | publisher=Cambridge University Press | url=http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20%281964%29%28L%29%28T%29%2859s%29.pdf | access-date=2016-07-23 | archive-url=https://web.archive.org/web/20170624164738/http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20(1964)(L)(T)(59s).pdf | archive-date=2017-06-24 | url-status=dead }}
• Beukers, Frits (2002), [https://webspace.science.uu.nl/~beuke106/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 | name-list-style= amp | last4= Tricomi | first4= Francesco G. | title= Higher transcendental functions | volume= 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 | access-date= 2011-07-30 | archive-date= 2011-08-11 | archive-url= https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf | url-status= dead }}
• 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}}.
• {{cite journal | last= Gauss | first= Carl Friedrich | author-link= Carl Friedrich Gauss | title= Disquisitiones generales circa seriem infinitam   $1\; +\; \backslash tfrac\; \{\backslash alpha\; \backslash beta\}\; \{1\; \backslash cdot\; \backslash gamma\}\; ~x\; +\; \backslash tfrac\; \{\backslash alpha\; (\backslash alpha+1)\; \backslash beta\; (\backslash beta+1)\}\; \{1\; \backslash cdot\; 2\; \backslash cdot\; \backslash gamma\; (\backslash gamma+1)\}\; ~x~x\; +\; \backslash mbox\{etc.\}$ | language= la | url= https://books.google.com/books?id=uDMAAAAAQAAJ | location= Göttingen | journal= Commentationes Societatis Regiae Scientarum Gottingensis Recentiores | year= 1813 | volume= 2 }}
• {{cite book | last1= Gelfand | first1= I. M. | author1-link=Israel Gelfand | last2= Gindikin | first2= S.G. | author2-link=Simon Gindikin | name-list-style= amp | last3= Graev | first3= M.I. | author3-link=Mark Iosifovich Graev | title= Selected topics in integral geometry | orig-year= 2000 | url= https://books.google.com/books?isbn=0821829327 | publisher= American Mathematical Society | location= Providence, R.I. | series= Translations of Mathematical Monographs | year= 2003 | volume= 220 | isbn= 978-0-8218-2932-5 | mr= 2000133 }}
• {{cite journal | last1= Gessel | first1= Ira |author-link=Ira Gessel | name-list-style= amp | 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 }}
• {{cite journal |last=Goursat |first=Édouard |author-link=Édouard Goursat |title=Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique |language=fr |url=http://www.numdam.org/item?id=ASENS_1881_2_10__S3_0 |access-date=2008-10-16 |journal=Annales Scientifiques de l'École Normale Supérieure |volume=10 |year=1881 |pages=3–142 |doi=10.24033/asens.207 |doi-access=free }}
• {{cite book | last1= Heckman | first1= Gerrit | last2= Schlichtkrull | first2= Henrik | name-list-style= amp | title= Harmonic Analysis and Special Functions on Symmetric Spaces | location= San Diego | publisher= Academic Press | year= 1994 | isbn= 0-12-336170-2 }} (part 1 treats hypergeometric functions on Lie groups)
• {{cite book | last=Hille | first=Einar | author-link=Einar Hille | 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 }}
• {{cite book|last=Ince|first=E. L.|author-link=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 | author-link=Felix Klein | title= Vorlesungen über die hypergeometrische Funktion | language= de | 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 }}
• {{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 | doi-access= free }}
• {{cite journal | last= Kummer | first= Ernst Eduard | title= Über die hypergeometrische Reihe $1\; +\; \backslash tfrac\; \{\backslash alpha\backslash cdot\; \backslash beta\}\; \{1\; \backslash cdot\; \backslash gamma\}\; ~x\; +\; \backslash tfrac\; \{\backslash alpha\; (\backslash alpha+1)\; \backslash beta\; (\backslash beta+1)\}\; \{1\; \backslash cdot\; 2\; \backslash cdot\; \backslash gamma\; (\backslash gamma+1)\}\; x^2\; +\; \backslash tfrac\; \{\backslash alpha\; (\backslash alpha+1)(\backslash alpha+2)\; \backslash beta\; (\backslash beta+1)(\backslash beta+2)\}\; \{1\; \backslash cdot\; 2\backslash cdot\; 3\; \backslash cdot\; \backslash gamma\; (\backslash gamma+1)(\backslash gamma+2)\}\; x^3\; +\backslash cdots$ | language= de | 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 }}
• {{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 ₃F₂ | journal = J. Comput. Appl. Math. | volume=72 | issue=2 | pages=293–300 | doi=10.1016/0377-0427(95)00279-0 | doi-access=free }}
• {{cite book | last1= Press | first1= W.H. | last2= Teukolsky | first2= S.A. | last3= Vetterling | first3= W.T. | last4= Flannery | first4= B.P. | name-list-style= amp | 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 }}
• {{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 | doi=10.4134/BKMS.2011.48.1.151 | doi-access=free }}
• {{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 }}
• {{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= de | volume= 7 | pages= 3–22 | publisher= Verlag der Dieterichschen Buchhandlung | location= Göttingen | url= http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002018691 }} (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 | author-link= 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 }}
• {{cite book | last= Slater | first= Lucy Joan |author-link=Lucy Joan Slater| title= Generalized hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 }} (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 | issue = 1–2 | pages = 473–487 | doi=10.1016/j.cam.2004.09.053 | arxiv = math/0310436 | bibcode = 2005JCoAM.178..473V | s2cid = 119596800 }}
• {{cite book | last= Wall | first= H.S. | title= Analytic Theory of Continued Fractions | publisher= D. Van Nostrand Company, Inc. | year= 1948 }}
• {{cite book | last1= Whittaker | first1= E.T. |author-link1=E. T. Whittaker | last2= Watson | first2= G.N.|author-link2=G. N. Watson | name-list-style= amp | title= A Course of Modern Analysis | location= Cambridge, UK | publisher= Cambridge University Press | year= 1927 }}
• {{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 }}

• {{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:Ordinary differential equations

Category:Series (mathematics)