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Hypergeometric series

In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. The series, if convergent, will define a hypergeometric function which may then be defined over a wider domain of the argument by analytic continuation. Hypergeometric functions generalize many special functions, including the Bessel functions, the incomplete Gamma function, the error function, the elliptic integrals and the orthogonal polynomials. This is in part because the hypergeometric functions are solutions to the hypergeometric differential equation, which is a fairly general second-order ordinary differential equation.

The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and less frequently studied series. The basic series is the q-analog of the ordinary hypergeometric series. There are several generalizations of the ordinary hypergeometric series, including a generalization to Riemann symmetric spaces.

The series 2F1

The classical standard hypergeometric series is given by:

$,_2F_1 \left(a,b;c;z\right) = sum_\left\{n=0\right\}^infty$
frac{(a)_n(b)_n}{(c)_n} , frac {z^n} {n!}

where (a)n = a(a+1)(a+2)…(a+n−1) is the rising factorial, or Pochhammer symbol. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence. This series is one of 24 closely related solutions, the Kummer solutions, of the hypergeometric differential equation.

By rewriting the series as


,_2F_1 (a,b;c;z) = 1 + sum_{n=1}^infty alpha_n z^n, ,

where the nth coefficient αn is given by


alpha_n = frac{(a)_n (b)_n}{(c)_n n!}, ,

it is easily seen that

• if either a or b is a negative integer, the series only has a finite number of terms;
• if c is a negative integer, the series is not well-defined because all the denominators are zero after a certain point;
• as n→∞, the ratio of successive coefficients αn + 1n approaches one in the limit; and
• the series is in general a convergent power series for values of z such that |z| < 1.

It can also be shown that if c is not a negative integer, the series converges when z = 1 if ℜ(cab) > 0. In this important case the value 2F1(a, b; c; 1) is given by


,_2F_1(a,b;c;1) = frac{Gamma(c) Gamma(c-a-b)}{Gamma(c-a) Gamma(c-b)} ,

where Γ(z) denotes the gamma function.

Elementary functions

Several of the familiar elementary functions can be expressed in terms of the hypergeometric function 2F1. These include


begin{align} (1+z)^k & = ,_2F_1left(-k,1;1;-zright)[3pt] sqrt{1-z^2} & = ,_2F_1left({scriptstylefrac{1}{2}},{scriptstyle -frac{1}{2}}; {scriptstylefrac{1}{2}};z^2right)[3pt] log(1+z) & = z ,_2F_1left(1,1;2;-zright)[3pt] logleft(frac{1+z}{1-z}right) & = 2z,_2F_1left({scriptstylefrac{1}{2}},1; {scriptstylefrac{3}{2}};z^2right)[3pt] arcsin(z) & = z ,_2F_1left({scriptstylefrac{1}{2}},{scriptstyle frac{1}{2}}; {scriptstylefrac{3}{2}};z^2right)[3pt] arctan(z) & = z ,_2F_1left({scriptstylefrac{1}{2}},1; {scriptstylefrac{3}{2}};-z^2right), end{align}

Special cases and applications

The classic orthogonal polynomials can all be expressed as special cases of $\left\{;\right\}_2F_1$ with one or both a and b being (negative) integers. Many other special cases are listed in the Special hypergeometric functions.

The function 2F1 has several integral representations, including the Euler hypergeometric integral.

Applications of hypergeometric series include the inversion of elliptic integrals; these are constructed by taking the ratio of the two linearly independent solutions of the hypergeometric differential equation to form Schwarz-Christoffel maps of the fundamental domain to the complex projective line or Riemann sphere.

A wide range of integrals of simple functions can be expressed using the hypergeometric function, e.g.:


int_0^xsqrt{1+y^alpha},mathrm{d}y = frac{xleft(alpha,{}_2F_1left(frac{1}{alpha},frac{1}{2};1+frac{1}{alpha};-x^alpharight)+2sqrt{x^alpha+1},right)}{2+alpha} , alphaneq0.

A limiting case of 2F1 is the Kummer function 1F1(a,b;z), known as the confluent hypergeometric function.

The series pFq

In the general case, the hypergeometric series is written as:

$,_pF_q\left(a_1,ldots,a_p;b_1,ldots,b_q;z\right)=sum_\left\{n=0\right\}^infty$
frac {alpha_n z^n}{n!},

where $alpha_0=1$ and

$frac\left\{alpha_\left\{n+1\right\}\right\}\left\{alpha_n\right\} = frac\left\{\left(n+a_1\right)\left(n+a_2\right)cdots\left(n+a_p\right)\right\}\left\{\left(n+b_1\right)\left(n+b_2\right)cdots\left(n+b_q\right)\right\}.$

The series may also be written:

$,_pF_q\left(a_1,ldots,a_p;b_1,ldots,b_q;z\right)=sum_\left\{n=0\right\}^infty$
frac{(a_1)_n(a_2)_nldots(a_p)_n}{(b_1)_n(b_2)_nldots(b_q)_n},frac{z^n}{n!},

where

$\left(a\right)_n=a\left(a+1\right)\left(a+2\right)...\left(a+n-1\right),$

is the rising factorial or Pochhammer symbol.

The series without the factor of n! in the denominator (summed over all integers n) is called the bilateral hypergeometric series.

Identities

A number of hypergeometric function identities were discovered in the nineteenth and twentieth centuries; one classical list of such identities is Bailey's list.

It is currently understood that there is a very large number of such identities, and several algorithms are now known to generate and prove these identities. Some mathematicians research on the various patterns that emerge from these algorithms.

Formal definition

A hypergeometric series is formally defined as any formal power series

$sum_n beta_n z^n$

in which the ratio of successive coefficients

$frac\left\{beta_\left\{n+1\right\}\right\}\left\{beta_n\right\}$

is a rational function of n. That is,

$frac\left\{beta_\left\{n+1\right\}\right\}\left\{beta_n\right\} = frac\left\{A\left(n\right)\right\}\left\{B\left(n\right)\right\}$

for some polynomials A(n) and B(n). Thus, for example, in the case of a geometric series, this ratio is a constant. Another example is the series for the exponential function, for which

$frac\left\{beta_\left\{n+1\right\}\right\}\left\{beta_n\right\} =frac\left\{1\right\}\left\{n+1\right\}.$

In practice the series is written as an exponential generating function, modifying the coefficients so that a general term of the series takes the form

$beta_n = frac\left\{alpha_n\right\}\left\{n!\right\},$

and $alpha_0=1$. One uses the exponential function as a 'baseline' for discussion.

Many interesting series in mathematics have the property that the ratio of successive terms is a rational function. However, when expressed as an exponential generating function, such series have a non-zero radius of convergence only under restricted conditions. Thus, by convention, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function with a non-zero radius of convergence. Such a function, and its analytic continuations, is called the hypergeometric function.

Convergence conditions were given by Carl Friedrich Gauss, who examined the case of

$frac\left\{alpha_\left\{n+1\right\}\right\}\left\{alpha_n\right\} = frac\left\{\left(n+a\right)\left(n+b\right)\right\}\left\{\left(n+c\right)\right\},$
leading to the classical standard hypergeometric series

$,_2F_1\left(a,b;c;z\right).$

Notation

The standard notation for the general hypergeometric series is

$,_mF_p.$

Here, the integers m and p refer to the degree of the polynomials P and Q, respectively, referring to the ratio

$frac\left\{alpha_\left\{n+1\right\}\right\}\left\{alpha_n\right\} = frac\left\{P\left(n\right)\right\}\left\{Q\left(n\right)\right\}.$

If m>p+1, the radius of convergence is zero and so there is no analytic function. The series naturally terminates in case P(n) is ever 0 for n a natural number. If Q(n) were ever zero, the coefficients would be undefined.

The full notation for F assumes that P and Q are monic and factorised, so that the notation for F includes an m-tuple that is the list of the negatives of the zeroes of P and a p-tuple of the negatives of the zeroes of Q. This is not much of a restriction: the fundamental theorem of algebra applies, and we can also absorb a leading coefficient of P or Q by redefining z. As a result of the factorisation, a general term in the series then takes the form of a ratio of products of Pochhammer symbols. Since Pochhammer notation for rising factorials is traditional it is neater to write F with the negatives of the zeros. Thus, to complete the notational example, one has

$,_2F_1 \left(a,b;c;z\right) = sum_\left\{n=0\right\}^infty$
frac{(a)_n(b)_n}{(c)_n} , frac {z^n} {n!} where $\left(a\right)_n=a\left(a+1\right)\left(a+2\right)...\left(a+n-1\right)$ is the rising factorial or Pochhammer symbol. Here, the zeros of P were −a and −b, while the zero of Q was −c.

History and generalizations

Studies in the nineteenth century included those of Ernst Kummer, and the fundamental characterisation by Bernhard Riemann of the F-function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for the 2F1, examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities: that effectively the entire algorithmic side of the theory was a consequence of basic facts and the use of Möbius transformations as a symmetry group.

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

Subsequently the hypergeometric series were generalised to several variables, for example by Paul Emile Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratio of successive terms, instead of being a rational function of n, are considered to be a rational function of $q^n$. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n.

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of hypergeometric series, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes).

Hypergeometric series can be developed on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through a special case: the hypergeometric series 2F1 is closely related to the Legendre polynomials, and when used in the form of spherical harmonics, it expresses, in a certain sense, the symmetry properties of the two-sphere or equivalently the rotations given by the Lie group SO(3). Concrete representations are analogous to the Clebsch-Gordan coefficients.