Generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation.
This series is usually denoted by or Using the rising factorial or Pochhammer symbol this can be written (Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)
The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion which could be written za−1e−z 2F0(1−a,1;;−z−1).
There are several such generalizations of the ordinary hypergeometric series, including the ones coming from zonal spherical functions on Riemannian symmetric spaces.
in the denominator (summed over all integers n, including negative) is called the bilateral hypergeometric series.
There are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0.
Excluding these cases, the ratio test can be applied to determine the radius of convergence.
and z is real, then the following convergence result holds Quigley et al. (2013): It is immediate from the definition that the order of the parameters aj, or the order of the parameters bk can be changed without changing the value of the function.
For example, This cancelling is a special case of a reduction formula that may be applied whenever a parameter on the top row differs from one on the bottom row by a non-negative integer.
[1][2] The following basic identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones[3] The generalized hypergeometric function satisfies and
Combining these gives a differential equation satisfied by w = pFq: Take the following operator: From the differentiation formulas given above, the linear space spanned by contains each of Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent: [4][5]
These dependencies can be written out to generate a large number of identities involving
For example, in the simplest non-trivial case, So This, and other important examples, can be used to generate continued fraction expressions known as Gauss's continued fraction.
A function obtained by adding ±1 to exactly one of the parameters aj, bk in is called contiguous to Using the technique outlined above, an identity relating
A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries.
A 20th century contribution to the methodology of proving these identities is the Egorychev method.
Terminating means that m is a non-negative integer and 2-balanced means that Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.
Identity 4. which is a finite sum if b-d is a non-negative integer.
Kummer's relation is Clausen's formula was used by de Branges to prove the Bieberbach conjecture.
(If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)
are called confluent hypergeometric functions of the first kind, also written
This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.
For each integer n≥2, the roots of the polynomial xn−x+t can be expressed as a sum of at most N−1 hypergeometric functions of type n+1Fn, which can always be reduced by eliminating at least one pair of a and b parameters.
[13] The generalized hypergeometric function is linked to the Meijer G-function and the MacRobert E-function.
Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and Joseph Kampé de Fériet; but a comparable general theory took long to emerge.
A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century.
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 general hypergeometric functions, 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).
Their importance and role can be understood through the following example: the hypergeometric series 2F1 has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO(3).
In tensor product decompositions of concrete representations of this group Clebsch–Gordan coefficients are met, which can be written as 3F2 hypergeometric series.
