So this satisfies the definition with A(n) = 1 and B(n) = n + 1.
It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers.
For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.
The ratio between consecutive coefficients now has the form
,
where c and d are the leading coefficients of A and B. The series then has the form
,
or, by scaling z by the appropriate factor and rearranging,
which could be written za−1e−z2F0(1−a,1;;−z−1). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function.
The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from zonal spherical functions on Riemannian symmetric spaces.
The series without the factor of n! in the denominator (summed over all integers n, including negative) is called the bilateral hypergeometric series.
Convergence conditions
There are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0.
If any aj is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree −aj.
If any bk is a non-positive integer (excepting the previous case with bk < aj) then the denominators become 0 and the series is undefined.
Excluding these cases, the ratio test can be applied to determine the radius of convergence.
If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z and thus defines an entire function of z. An example is the power series for the exponential function.
If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 and diverges for |z| > 1. Whether it converges for |z| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of z.
If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally.
The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z = 1 if
.
Further, if p=q+1, and z is real, then the following convergence result holds Quigley et al. (2013):
.
Basic properties
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. Also, if any of the parameters aj is equal to any of the parameters bk, then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. 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]
Euler's integral transform
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]
Differentiation
The generalized hypergeometric function satisfies
and
Additionally,
Combining these gives a differential equation satisfied by w = pFq:
.
Contiguous function and related identities
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 .
Similarly, by applying the differentiation formulas twice, there are such functions contained in
which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way.
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 and its two contiguous functions can be given, six identities relating and any two of its four contiguous functions, and fifteen identities relating and any two of its six contiguous functions have been found. (The first one was derived in the previous paragraph. The last fifteen were given by Gauss in his 1812 paper.)
Identities
For identities involving the Gauss hypergeometric function 2F1, see Hypergeometric function.
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.
The functions of the form are called confluent hypergeometric functions of the first kind, also written . The incomplete gamma function is a special case.
The differential equation for this function is
or
When b is not a positive integer, the substitution
gives a linearly independent solution
so the general solution is
where k, l are constants.
When a is a non-positive integer, −n, is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of 1F1 as well.
The series 1F2
Relations to other functions are known for certain parameter combinations only.
The function is the antiderivative of the cardinal sine. With modified values of and , one obtains the antiderivative of .[8]
Historically, the most important are the functions of the form . These are sometimes called Gauss's hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function is used for the functions pFq if there is risk of confusion. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.
where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, known as the Kummer solutions, derivable using various identities, valid in different regions of the complex plane.
When a is a non-positive integer, −n,
is a polynomial. Up to constant factors and scaling, these are the Jacobi polynomials. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2F1 as well. This includes Legendre polynomials and Chebyshev polynomials.
A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.:
All roots of a quintic equation can be expressed in terms of radicals and the Bring radical, which is the real solution to . The Bring radical can be written as:[13]
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]
Generalizations
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. 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 ratios considered of successive terms, instead of a rational function of n, are a rational function of qn. 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).
Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups. 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.
Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.
Fox–Wright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.
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Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. London: Cambridge University Press. Zbl0011.02303.
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