In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by is a rational function of . The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series.
Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as Bessel functions, and the classical orthogonal polynomials.
Notation A hypergeometric series is formally defined as a power series
in which the ratio of successive coefficients is a rational function of the index ; that is,
where and are polynomials in . For example, in the case of the series for the exponential function
we have
This satisfies the definition with and .
It is customary to factor out the leading term, so is assumed to be 1. The polynomials can be factored into linear factors of the form and respectively, where the and are complex numbers.
For historical reasons, it is assumed that is a factor of . If this is not already the case then both and 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 and are the leading coefficients of and . The series then has the form , or, by scaling by the appropriate factor and rearranging, .
This has the form of an exponential generating function. This series is usually denoted by
We can then use the rising factorial, typically written with a Pochhammer symbol in this context (even though the Pochhammer symbol normally denotes the falling factorial), defined for all by and
where represents the gamma function. The series can then be written using either the rising factorial or the gamma function as
Terminology When all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines an analytic function. Such a function, and its analytic continuations, is called the hypergeometric function.
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 can be written as . 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 in the denominator (summed over all integers , including negative) is called the bilateral hypergeometric series.
Convergence conditions There are certain values of the parameters and for which the numerator or denominator of the coefficients is 0. If any 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 . If any is a non-positive integer (excepting the previous case with ) 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 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of and thus defines an entire function of . An example is the power series for the exponential function. If then the ratio of coefficients tends to one. This implies that the series converges for and diverges for . Whether it converges for is more difficult to determine. Analytic continuation can be employed for larger values of . If then the ratio of coefficients grows without bound. This implies that, besides , 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 when is on the unit circle is more difficult. It can be shown that the series converges absolutely at if . Furthermore, if , then and is real, and the following convergence result holds: .
Basic properties It is immediate from the definition that the order of the parameters , or the order of the parameters can be changed without changing the value of the function. Also, if any of the parameters is equal to any of the parameters , 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:
Euler's integral transform The following identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones:
Differentiation The generalized hypergeometric function satisfies
and
for . Additionally,
Combining these gives a differential equation satisfied by : .
Contiguous function and related identities Consider the operator
From the differentiation formulas given above, the linear space spanned by
contains each of
Since this space is two-dimensional, any three of these functions are linearly dependent:
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.
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 to exactly one of the parameters , 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.
Identities
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.
Saalschütz's theorem Saalschütz's theoremSee or for a proof, or the ProofWiki. states
For extension of this theorem, see a research paper by Rakha & Rathie. According to , it was in fact first discovered by Pfaff in 1797.Pfaff, J. F. . Observations analyticae ad L. Euleri Institutiones Calculi Integralis. Vol. IV, Supplem. II et IV, Historie de 1793, Nova Acata Acad. Scie. Petropolitanae. XI, 38-57. (Note: The history section is paged separately from the scientific section of this journal.)
Dixon's identity
Dixon's identity,See for a detailed proof. An alternative proof is in first proved by , gives the sum of a well-poised 3F2 at 1:
For generalization of Dixon's identity, see a paper by Lavoie, et al.
Dougall's formula Dougall's formula gives the sum of a very well-poised series that is terminating and 2-balanced.
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. It is also called the Dougall-Ramanujan identity. It is a special case of Jackson's identity, and it gives Dixon's identity and Saalschütz's theorem as special cases.
Generalization of Kummer's transformations and identities for 2F2 Identity 1.
where ;
Identity 2.
which links Bessel functions to 2F2; this reduces to Kummer's second formula for b = 2a:
Identity 3. .
Identity 4.
which is a finite sum if b-d is a non-negative integer.
Kummer's relation Kummer's relation is
Clausen's formula
Clausen's formula
was used by de Branges to prove the Bieberbach conjecture.
Special cases Many of the special functions in mathematics are special cases of the confluent hypergeometric function or the hypergeometric function; see the corresponding articles for examples.
The series 0F0
As noted earlier, . The differential equation for this function is , which has solutions where k is a constant.
The series 0F1 The functions of the form are called confluent hypergeometric limit functions and are closely related to Bessel functions.
The relationship is:
The differential equation for this function is
or
When a is not a positive integer, the substitution
gives a linearly independent solution
so the general solution is
where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)
A special case is:
The series 1F0
An important case is:
The differential equation for this function is
or
which has solutions
where k is a constant.
is the geometric series with ratio z and coefficient 1.
is also useful.
The series 1F1
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 .Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)
The Lommel function is .Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)
The series 2F0 The confluent hypergeometric function of the second kind can be written as:
The series 2F1
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.
The differential equation for this function is
or
It is known as the hypergeometric differential equation. When c is not a positive integer, the substitution
gives a linearly independent solution
so the general solution for |z| < 1 is
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,