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In mathematics, the Gaussian or ordinary hypergeometric function _{2}F_{1}(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. [It] is a solution of a secondorder linear ordinary differential equation (ODE). Every secondorder linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1953), Abramowitz & Stegun (1965), and Daalhuis (2010).
History
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813). Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the secondorder differential equation for _{2}F_{1}(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.
The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).
The hypergeometric series
The hypergeometric function is defined for z < 1 by the power series
 $\{\}\_2F\_1(a,b;c;z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(a)\_n\; (b)\_n\}\{(c)\_n\}\; \backslash frac\{z^n\}\{n!\}.$
It is undefined (or infinite) if c equals a nonpositive integer. Here (q)_{n} is the (rising) Pochhammer symbol, which is defined by:
 $(q)\_n\; =\; \backslash begin\{cases\}\; 1\; \&\; n\; =\; 0\; \backslash \backslash $
q(q+1) \cdots (q+n1) & n > 0.
\end{cases}
The series terminates if either a or b is a nonpositive integer. For complex arguments z with z ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1.
As c goes to a nonpositive integer −m, _{2}F_{1}(z) goes to infinity, but if we divide by the gamma function Γ(c), we have a limit:
$\backslash lim\_\{c\backslash rightarrow\; m\}\backslash frac\{(m+1)!\}z^\{m+1\}\{\}\_2F\_1(a+m+1,b+m+1;m+2;z)$
_{2}F_{1}(z) is the most usual type of generalized hypergeometric series _{p}F_{q}, and is often designated simply F(z).
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
 $\backslash begin\{align\}$
\ln(1+z) &= z\, _2F_1(1,1;2;z) \\
(1z)^{a} &= \, _2F_1(a,1;1;z) \\
\arcsin(z) &= z \, _2F_1\left(\tfrac{1}{2}, \tfrac{1}{2}; \tfrac{3}{2};z^2\right).
\end{align}
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
 $\{\}\_2F\_1(a,1a;c;z)\; =\; \backslash Gamma(c)z^\{\backslash tfrac\{1c\}\{2\}\}(1z)^\{\backslash tfrac\{c1\}\{2\}\}P\_\{a\}^\{1c\}(12z)$
Several orthogonal polynomials, including Jacobi polynomials P(α,β)
n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using
 $\{\}\_2F\_1(n,\backslash alpha+1+\backslash beta+n;\backslash alpha+1;x)\; =\; \backslash frac\{n!\}\{(\backslash alpha+1)\_n\}P^\{(\backslash alpha,\backslash beta)\}\_n(12x)$
Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.
Elliptic modular functions 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
 $\backslash tau\; =\; \{\backslash rm\{i\}\}\backslash frac\}\}\}$
Transformation formulas
Transformation formulas relate two hypergeometric functions at different values of the argument z.
Fractional linear transformations
Euler's transformation is
 $\{\}\_2F\_1\; (a,b;c;z)\; =\; (1z)^\{cab\}\; \{\}\_2F\_1\; (ca,\; cb;c;\; z).$
It follows by combining the two Pfaff transformations
 $\{\}\_2F\_1\; (a,b;c;z)\; =\; (1z)^\{b\}\; \{\}\_2F\_1\; \backslash left\; (b,ca;c;\backslash tfrac\{z\}\{z1\}\; \backslash right\; )$
 $\{\}\_2F\_1\; (a,b;c;z)\; =\; (1z)^\{a\}\; \{\}\_2F\_1\; \backslash left\; (a,\; cb;c;\; \backslash tfrac\{z\}\{z1\}\; \backslash right\; )$
which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see papers by Rathie & Paris and Rakha & Rathie.
Quadratic transformations
If two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by Kummer (1836), and a complete list was given by Goursat (1881). A typical example is
 $F(a,b;2b;z)\; =\; (1z)^\{\backslash frac\{a\}\{2\}\}\; F\; \backslash left\; (\backslash tfrac\{1\}\{2\}a,\; b\backslash tfrac\{1\}\{2\}a;\; b+\backslash tfrac\{1\}\{2\};\; \backslash frac\{z^2\}\{4z4\}\; \backslash right)$
Higher order transformations
If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by Goursat (1881). A typical example is
 $F\; \backslash left\; (\backslash tfrac\{3\}\{2\}a,\backslash tfrac\{1\}\{2\}(3a1);a+\backslash tfrac\{1\}\{2\};\backslash tfrac\{z^2\}\{3\}\; \backslash right)\; =\; (1+z)^\{13a\}F\; \backslash left\; (a\backslash tfrac\{1\}\{3\},\; a,\; 2a,\; 2z(3+z^2)(1+z)^\{3\}\; \backslash right\; )$
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.
Values at special points z
See (Slater 1966, Appendix III) for a list of summation formulas at special points, most of which also appear in (Bailey 1935). (Gessel & Stanton 1982) gives further evaluations at more points. (Koepf 1995) shows how most of these identities can be verified by computer algorithms.
Special values at z = 1
Gauss's theorem, named for Carl Friedrich Gauss, is the identity
 $\{\}\_2F\_1\; (a,b;c;1)=\; \backslash frac\{\backslash Gamma(c)\backslash Gamma(cab)\}\{\backslash Gamma(ca)\backslash Gamma(cb)\},\; \backslash qquad\; \backslash Re(c)>\backslash Re(a+b)$
which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity, first found by Zhu Shijie (= Chu ShiChieh), as a special case.
Dougall's formula generalizes this to the bilateral hypergeometric series at z = 1.
Kummer's theorem (z = −1)
There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformation to change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer:
 $\{\}\_2F\_1\; (a,b;1+ab;1)=\; \backslash frac\{\backslash Gamma(1+ab)\backslash Gamma(1+\backslash tfrac12a)\}\{\backslash Gamma(1+a)\backslash Gamma(1+\backslash tfrac12ab)\}$
which follows from Kummer's quadratic transformations
 $\backslash begin\{align\}$
_2F_1(a,b;1+ab;z)&= (1z)^{a} \;_2F_1 \left(\frac a 2, \frac{1+a}2b; 1+ab; \frac{4z}{(1z)^2}\right)\\
&=(1+z)^{a} \, _2F_1\left(\frac a 2, \frac{a+1}2; 1+ab; \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 a paper by Lavoie, et al.
Values at z = 1/2
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
 $\_2F\_1\; \backslash left(a,1a;c;\backslash tfrac12\backslash right)=\; \backslash frac\{\backslash Gamma(\backslash tfrac12c)\backslash Gamma(\backslash tfrac12\backslash left(1+c\backslash right))\}\{\backslash Gamma(\backslash tfrac12\backslash left(c+a\backslash right))\backslash Gamma(\backslash tfrac12\backslash left(1+ca\backslash right))\}.$
For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see a paper by Lavoie, et al.
Other points
There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in
(Gessel & Stanton 1982) and (Koepf 1995). Some typical examples are given by
 $\{\}\_2F\_1\; \backslash left(a,a;\backslash tfrac\{1\}\{2\};\backslash tfrac\{1x\}\{2\}\; \backslash right\; )\; =\; \backslash frac\{(1x)^a+(1x)^\{a\}\}\{2\},$
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.
See also
 Appell series, a 2variable generalization of hypergeometric series
 Basic hypergeometric series where the ratio of terms is a periodic function of the index
 Bilateral hypergeometric series _{p}H_{p} are similar to generalized hypergeometric series, but summed over all integers
 Binomial series _{1}F_{0}
 Confluent hypergeometric series _{1}F_{1}(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 _{2}F_{1}
 Fox Hfunction, an extension of the Meijer Gfunction
 Fox–Wright function, a generalization of the generalized hypergeometric function
 Frobenius solution to the hypergeometric equation
 General hypergeometric function introduced by Gelfand.
 Generalized hypergeometric series _{p}F_{q} 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 differential equation, a secondorder linear ordinary differential equation
 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 Efunction, an extension of the generalized hypergeometric series _{p}F_{q} to the case p>q+1.
 Meijer Gfunction, an extension of the generalized hypergeometric series _{p}F_{q} 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
References
 .

 Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge.
 Beukers, Frits (2002), Gauss' hypergeometric function. (lecture notes reviewing basics, as well as triangle maps and monodromy)


 Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0521833574.
 (a reprint of this paper can be found in , p. 125)



 (part 1 treats hypergeometric functions on Lie groups)



 Lavoie,J.L., Grondin, F. and Rathie, A.K., Generalizations of Whipple's theorem on the sum of a _{3}F_{2}, J. Comput. Appl. Math., 72, 293300,(1996).

 (a reprint of this paper can be found in All publications of Riemann PDF)

 (there is a 2008 paperback with ISBN 9780521090612)



 Rathie, Arjun K. & Paris, R.B.: An extension of the Euler'stype transformation for the 3F2 series: Far East J.Math.Sci., 27(1), 4348 (2007).
 Rakha, M.A. & Rathie, Arjun K. : Extensions of Euler's type II transformation and Saalschutz's theorem: Bull. Korean Math. Soc.,48(1), 151156 (2011).
External links
 Template:Springer
 John Pearson, University of Oxford, MSc Thesis)
 Marko Petkovsek, Herbert Wilf and Doron Zeilberger, The book "A = B" (freely downloadable)
 MathWorld.
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