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Hypergeometric function

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Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1953), Abramowitz & Stegun (1965), and Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.

History

The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.

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

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

Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.

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

The hypergeometric series

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

{}_2F_1(q,b;c;z) = \sum_{n=0}^\infty \frac{(q)_n (b)_n}{(c)_n} \frac{z^n}{n!}.

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

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

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

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

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 → −m, where m is a positive integer, 2F1(z) → ∞, but if we divide by Γ(c), we have a limit:

\lim_{c\to -m}\frac{(m+1)!}z^{m+1}{}_2F_1(q+m+1,b+m+1;m+2;z)

2F1(z) is the most usual type of generalized hypergeometric series pFq, 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

\ln(1+z) = z\ _2F_1(1,1;2;-z)
(1-z)^{-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)

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

M(a,c,z) = \lim_{b\to \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) = \Gamma(c)z^{\tfrac{1-c}{2}}(1-z)^{\tfrac{c-1}{2}}P_{-a}^{1-c}(1-2z)

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

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

\tau = {\rm{i}}\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) = (1-z)^{c-a-b} {}_2F_1 (c-a, c-b;c ; z).

It follows by combining the two Pfaff transformations

{}_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 )

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.

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

Higher order transformations

If 1−c, ab, a+bc 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 \left (\tfrac{3}{2}a,\tfrac{1}{2}(3a-1);a+\tfrac{1}{2};-\tfrac{z^2}{3} \right) = (1+z)^{1-3a}F \left (a-\tfrac{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 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

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)=\frac{ (c-b)_{(m)} }{(c)_{(m)} }

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+a-b;-1)= \frac{\Gamma(1+a-b)\Gamma(1+\tfrac12a)}{\Gamma(1+a)\Gamma(1+\tfrac12a-b)}

which follows from Kummer's quadratic transformations

\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 a paper by Lavoie, et al.

Values at z = 1/2

Gauss's second summation theorem is

_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))}.

Bailey's theorem is

_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))}.

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

which can be restated as

T_a(\cos x)={}_2F_1\left(a,-a;\tfrac{1}{2};\tfrac{1}{2}(1-\cos x)\right)=\cos(a x)

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

References

1. ^ Hille, Einar (1976), Ordinary differential equations in the complex domain, Dover, pp. 374–401, ISBN 0-486-69620-0, Chapter 10, "The Schwarzian".
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