Hypergeometric series

From Academic Kids

In mathematics, a hypergeometric series is the sum of a sequence of terms 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. The hypergeometric series is generally written:

<math>\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x)=\sum_{k=0}^\infty c_k x^k<math>

where <math>c_0<math>=1 and

<math>\frac{c_{k+1}}{c_k}=\frac{(k+a_1)(k+a_2)\cdots(k+a_p)}{(k+b_1)(k+b_2)\cdots(k+b_q)}\,\frac{1}{k+1}<math>

The series may also be written:

<math>\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x)=\sum_{k=0}^\infty

\frac{(a_1)_k(a_2)_k\ldots(a_p)_k}{(b_1)_k(b_1)_k\ldots(b_q)_k}\,\frac{x^k}{k!}<math>

where <math>(a)_k=a(a+1)\ldots(a+k-1)<math> is the rising factorial or Pochhammer symbol

Contents

Introduction

A hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients <math>\alpha_n/\alpha_{n-1}<math> is a rational function of n. That is,

<math>\frac{\alpha_n}{\alpha_{n-1}} = \frac{\tilde P(n)}{\tilde Q(n)}<math>

for some polynomials <math>\tilde P(n)<math> and <math>\tilde Q(n)<math>. 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

<math>\alpha_n/\alpha_{n-1}=z/n\,.<math>

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

<math>\alpha_n = \beta_n z^n /n!\,<math>

and <math>\beta_0=1<math>. Notice that z corresponds to a constant in the ratio <math>\tilde P/\tilde Q<math>. 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

<math>\frac{\beta_n}{\beta_{n-1}} = \frac{(n+a)(n+b)}{(n+c)}<math>,

leading to the classical standard hypergeometric series

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

Notation

The standard notation for the general hypergeometric series is

<math>\,_mF_p.<math>

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

<math>\frac{\beta_n}{\beta_{n-1}} = \frac{P(n)}{Q(n)}.<math>

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 zeroes of P and a p-tuple of the zeroes of Q. Note that this is not much 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

<math> \,_2F_1 (a,b;c;z) = \sum_{n=0}^\infty

\frac{(a)_n(b)_n}{(c)_n} \, \frac {z^n} {n!} <math> where <math>(a)_n=a(a+1)(a+2)...(a+n-1)<math> is the rising factorial or Pochhammer symbol. Here, the zeros of P were −a and −b, while the zero of Q was −c.

Special cases and applications

The classic orthogonal polynomials can all be expressed as special cases of <math>{\;}_2F_1<math> with one or both a and b being (negative) integers. Similarly, the Legendre functions are a special case as well.

Applications of hypergeometric series includes 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.

The Kummer function 1F1(a,b;x) is known as the confluent hypergeometric function.

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

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 F, 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.

Subsequently the hypergeometric series were generalised to several variables, for example by Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. What are called q-series analogues were found. 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, Gel'fand 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.

See also: hypergeometric function identities.

References

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