Generalized Hypergeometric Function
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In mathematics, a generalized hypergeometric series is a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
in which the ratio of successive
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s indexed by ''n'' is a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
. 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 In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
as special cases, which in turn have many particular
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined ...
as special cases, such as elementary functions,
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s, and the classical orthogonal polynomials.


Notation

A hypergeometric series is formally defined as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
:\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of ''n''. That is, :\frac = \frac where ''A''(''n'') and ''B''(''n'') are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s in ''n''. For example, in the case of the series for the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, :1+\frac+\frac+\frac+\cdots, we have: : \beta_n = \frac, \qquad \frac = \frac. So this satisfies the definition with and . 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 (''b''''k'' + ''n'') respectively, where the ''a''''j'' and ''b''''k'' are
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. 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 :\frac, where ''c'' and ''d'' are the leading coefficients of ''A'' and ''B''. The series then has the form :1 + \frac\frac + \frac \frac\left(\frac\right)^2+\cdots, or, by scaling ''z'' by the appropriate factor and rearranging, :1 + \frac\frac + \frac\frac+\cdots. This has the form of an
exponential generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
. This series is usually denoted by :_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) or :\,_pF_q \left begin a_1 & a_2 & \cdots & a_ \\ b_1 & b_2 & \cdots & b_q \end ; z \right Using the rising factorial or Pochhammer symbol :\begin (a)_0 &= 1, \\ (a)_n &= a(a+1)(a+2) \cdots (a+n-1), && n \geq 1 \end this can be written :\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_^\infty \frac \, \frac . (Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)


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 In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
. Such a function, and its
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
s, 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 :\Gamma(a,z) \sim z^e^\left(1+\frac+\frac + \cdots\right) which could be written ''z''''a''−1''e''−z 2''F''0(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 ''a''''j'' and ''b''''k'' for which the numerator or the denominator of the coefficients is 0. * If any ''a''''j'' 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 −''a''''j''. * If any ''b''''k'' is a non-positive integer (excepting the previous case with −''b''''k'' < ''a''''j'') 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 :\Re\left(\sum b_k - \sum a_j\right)>0. Further, if ''p''=''q''+1, \sum_^a_\geq\sum_^b_ and ''z'' is real, then the following convergence result holds : :\lim_(1-z)\frac=\sum_^a_-\sum_^b_.


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, :\,_2F_1(3,1;1;z) = \,_2F_1(1,3;1;z) = \,_1F_0(3;;z). 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. : _F_\left \begin a_,\ldots ,a_,c + n \\ b_,\ldots ,b_,c \end ;z\right= \sum_^n \binom \frac \frac _AF_B\left \begin a_ + j,\ldots ,a_ + j \\ b_ + j,\ldots ,b_ + j \end ;z\right


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 : _F_\left \begin a_,\ldots ,a_,c \\ b_,\ldots ,b_,d \end ;z\right=\frac \int_^t^(1-t)_^\ _F_\left \begin a_,\ldots ,a_ \\ b_,\ldots ,b_ \end ; tz\right dt


Differentiation

The generalized hypergeometric function satisfies :\begin \left (z\frac + a_j \right )_pF_q\left \begin a_1,\dots,a_j,\dots,a_p \\ b_1,\dots,b_q\end ;z\right&= a_j \; _pF_q\left \begin a_1,\dots,a_j+1,\dots,a_p \\ b_1,\dots,b_q \end ;z\right\\ \end and \begin \left (z\frac + b_k - 1 \right )_pF_q\left \begin a_1,\dots,a_p \\ b_1,\dots,b_k,\dots,b_q\end ;z\right&= (b_k - 1) \; _pF_q\left \begin a_1,\dots,a_p \\ b_1,\dots,b_k-1,\dots,b_q \end ;z \right\text b_k\neq 1 \end Additionally, \begin \frac \; _pF_q\left \begin a_1,\dots,a_p \\ b_1,\dots,b_q \end ;z \right&= \frac\; _pF_q\left \begin a_1+1,\dots,a_p+1 \\ b_1+1,\dots,b_q+1 \end ;z \right\end Combining these gives a differential equation satisfied by ''w'' = p''F''q: :z\prod_^\left(z\frac + a_n\right)w = z\frac\prod_^\left(z\frac + b_n-1\right)w.


Contiguous function and related identities

Take the following operator: :\vartheta = z\frac. From the differentiation formulas given above, the linear space spanned by :_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z), \vartheta\; _pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) contains each of :_pF_q(a_1,\dots,a_j+1,\dots,a_p;b_1,\dots,b_q;z), :_pF_q(a_1,\dots,a_p;b_1,\dots,b_k-1,\dots,b_q;z), :z\; _pF_q(a_1+1,\dots,a_p+1;b_1+1,\dots,b_q+1;z), :_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z). Since the space has dimension 2, any three of these ''p''+''q''+2 functions are linearly dependent. These dependencies can be written out to generate a large number of identities involving _pF_q. For example, in the simplest non-trivial case, : \; _0F_1(;a;z) = (1) \; _0F_1(;a;z), : \; _0F_1(;a-1;z) = (\frac+1) \; _0F_1(;a;z), :z \; _0F_1(;a+1;z) = (a\vartheta) \; _0F_1(;a;z), So : \; _0F_1(;a-1;z)- \; _0F_1(;a;z) = \frac \; _0F_1(;a+1;z). This, and other important examples, : \; _1F_1(a+1;b;z)- \, _1F_1(a;b;z) = \frac \; _1F_1(a+1;b+1;z), : \; _1F_1(a;b-1;z)- \, _1F_1(a;b;z) = \frac \; _1F_1(a+1;b+1;z), : \; _1F_1(a;b-1;z)- \, _1F_1(a+1;b;z) = \frac \; _1F_1(a+1;b+1;z) : \; _2F_1(a+1,b;c;z)- \, _2F_1(a,b;c;z) = \frac \; _2F_1(a+1,b+1;c+1;z), : \; _2F_1(a+1,b;c;z)- \, _2F_1(a,b+1;c;z) = \frac \; _2F_1(a+1,b+1;c+1;z), : \; _2F_1(a,b;c-1;z)- \, _2F_1(a+1,b;c;z) = \frac \; _2F_1(a+1,b+1;c+1;z), can be used to generate
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
expressions known as Gauss's continued fraction. Similarly, by applying the differentiation formulas twice, there are \binom such functions contained in :\\; _p F_q (a_1,\dots,a_p;b_1,\dots,b_q;z), 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 ''a''''j'', ''b''''k'' in :_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) is called contiguous to :_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z). Using the technique outlined above, an identity relating _0F_1(;a;z) and its two contiguous functions can be given, six identities relating _1F_1(a;b;z) and any two of its four contiguous functions, and fifteen identities relating _2F_1(a,b;c;z) 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

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 theorem is :_3F_2 (a,b, -n;c, 1+a+b-c-n;1)= \frac. For extension of this theorem, see a research paper by Rakha & Rathie.


Dixon's identity

Dixon's identity, first proved by , gives the sum of a well-poised 3''F''2 at 1: :_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac. 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. :\begin _7F_6 & \left(\begina&1+\frac&b&c&d&e&-m\\&\frac&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\ \end;1\right) = \\ &=\frac. \end Terminating means that ''m'' is a non-negative integer and 2-balanced means that :1+2a=b+c+d+e-m. Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.


Generalization of Kummer's transformations and identities for 2''F''2

Identity 1. :e^ \; _2F_2(a,1+d;c,d;x)= _2F_2(c-a-1,f+1;c,f;-x) where :f=\frac; Identity 2. :e^ \, _2F_2 \left(a, 1+b; 2a+1, b; x\right)= _0F_1 \left(;a+\tfrac; \tfrac \right) - \frac\; _0F_1 \left(;a+\tfrac 3 2; \tfrac \right), which links
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s to 2''F''2; this reduces to Kummer's second formula for ''b'' = 2''a'': Identity 3. :e^ \, _1F_1(a,2a,x)= _0F_1 \left (;a+\tfrac 1 2; \tfrac \right ). Identity 4. :\begin _2F_2(a,b;c,d;x)=& \sum_ \frac \; _1F_1(a+i;c+i;x)\frac \\ =& e^x \sum_ \frac \; _1F_1(c-a;c+i;-x)\frac, \end which is a finite sum if ''b-d'' is a non-negative integer.


Kummer's relation

Kummer's relation is :_2F_1\left(2a,2b;a+b+\tfrac 1 2;x\right)= _2F_1\left(a,b; a+b+\tfrac 1 2; 4x(1-x)\right).


Clausen's formula

Clausen's formula :_3F_2(2c-2s-1, 2s, c-\tfrac 1 2; 2c-1, c; x)=\, _2F_1(c-s-\tfrac 1 2,s; c; x)^2 was used by de Branges to prove the
Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...
.


Special cases

Many of the special functions in mathematics are special cases of the
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
or the hypergeometric function; see the corresponding articles for examples.


The series 0''F''0

As noted earlier, _0F_0(;;z) = e^z. The differential equation for this function is \fracw = w, which has solutions w = ke^z where ''k'' is a constant.


The series 1''F''0

An important case is: :_1F_0(a;;z) = (1-z)^. The differential equation for this function is :\fracw =\left (z\frac+a \right )w, or :(1-z)\frac = aw, which has solutions :w=k(1-z)^ where ''k'' is a constant. :_1F_0(1;;z) = \sum_ z^n = (1-z)^ is the
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...
with ratio ''z'' and coefficient 1. :z ~ _1F_0(2;;z) = \sum_ n z^n = z (1-z)^ is also useful.


The series 0''F''1

A special case is: :_0F_1\left(;\frac;-\frac\right)=\cos z


Example

We can get this result, using the formula with rising factorials, as follows: \begin _0F_1\left(;\tfrac;-\tfrac\right) &=\sum_^\infty \frac \, \frac =\sum_^\infty \frac \, \frac =\sum_^\infty \frac \\ &=\sum_^\infty \frac =\sum_^\infty \frac =\sum_^\infty \frac \\ &=\sum_^\infty \frac =\sum_^\infty\frac=\cos z \end The functions of the form _0F_1(;a;z) are called confluent hypergeometric limit functions and are closely related to
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s. The relationship is: :J_\alpha(x)=\frac _0F_1\left (;\alpha+1; -\tfracx^2 \right ). :I_\alpha(x)=\frac _0F_1\left (;\alpha+1; \tfracx^2 \right ). The differential equation for this function is :w = \left (z\frac+a \right )\frac or :z\frac+a\frac-w = 0. When ''a'' is not a positive integer, the substitution :w = z^u, gives a linearly independent solution :z^\;_0F_1(;2-a;z), so the general solution is :k\;_0F_1(;a;z)+l z^\;_0F_1(;2-a;z) 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.)


The series 1''F''1

The functions of the form _1F_1(a;b;z) are called confluent hypergeometric functions of the first kind, also written M(a;b;z). The incomplete gamma function \gamma(a,z) is a special case. The differential equation for this function is :\left (z\frac+a \right )w = \left (z\frac+b \right )\frac or :z\frac+(b-z)\frac-aw = 0. When ''b'' is not a positive integer, the substitution :w = z^u, gives a linearly independent solution :z^\;_1F_1(1+a-b;2-b;z), so the general solution is :k\;_1F_1(a;b;z)+l z^\;_1F_1(1+a-b;2-b;z) where ''k'', ''l'' are constants. When a is a non-positive integer, −''n'', _1F_1(-n;b;z) is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of 1''F''1 as well.


The series 2''F''0

This occurs in connection with the exponential integral function Ei(''z'').


The series 2''F''1

Historically, the most important are the functions of the form _2F_1(a,b;c;z). 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 ''p''''F''''q'' if there is risk of confusion. This function was first studied in detail by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
, who explored the conditions for its convergence. The differential equation for this function is : \left (z\frac+a \right ) \left (z\frac+b \right )w =\left (z\frac+c \right )\frac or :z(1-z)\frac + \left -(a+b+1)z \right\frac - ab\,w = 0. It is known as the hypergeometric differential equation. When ''c'' is not a positive integer, the substitution :w = z^u gives a linearly independent solution : z^\; _2F_1(1+a-c,1+b-c;2-c;z), so the general solution for , ''z'', < 1 is :k\; _2F_1(a,b;c;z)+l z^\; _2F_1(1+a-c,1+b-c;2-c;z) 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 Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist *Clare Kummer (1873—1958), American composer, lyricist and playwright *Clarence Kummer (1899–1930), American jockey * Christo ...
solutions, derivable using various identities, valid in different regions of the complex plane. When ''a'' is a non-positive integer, −''n'', :_2F_1(-n,b;c;z) 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 2''F''1 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.: :\int_0^x\sqrt\,\mathrmy=\frac\left \,\qquad \alpha\neq0.


The series 3''F''0

This occurs in connection with
Mott polynomials In mathematics the Mott polynomials ''s'n''(''x'') are polynomials introduced by who applied them to a problem in the theory of electrons. They are given by the exponential generating function : e^=\sum_n s_n(x) t^n/n!. Because the factor in t ...
.


The series 3''F''1

This occurs in the theory of Bessel functions. It provides a way to compute Bessel functions of large arguments.


Dilogarithm

::\operatorname_2(x) = \sum_\, = x \; _3F_2(1,1,1;2,2;x) is the dilogarithm


Hahn polynomials

::Q_n(x;a,b,N)= _3F_2(-n,-x,n+a+b+1;a+1,-N+1;1) is a Hahn polynomial.


Wilson polynomials

::p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \; _4F_3\left( \begin -n&a+b+c+d+n-1&a-t&a+t \\ a+b&a+c&a+d \end ;1\right) is a
Wilson polynomial In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the ...
.


Generalizations

The generalized hypergeometric function is linked to the Meijer G-function and the
MacRobert E-function In mathematics, the E-function was introduced by to extend the generalized hypergeometric series ''p'F'q''(·) to the case ''p'' > ''q'' + 1. The underlying objective was to define a very general function that includes as particular cases th ...
. Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and
Joseph Kampé de Fériet Marie-Joseph Kampé de Fériet ( Paris, 14 May 1893 – Villeneuve d'Ascq, 6 April 1982) was professor at Université Lille Nord de France from 1919 to 1969. Besides his works on mathematics and fluid mechanics, he directed the ''Institut de ...
; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the
q-series In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer sy ...
analogues, called the basic hypergeometric series, were given by
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Le ...
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 In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the 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 function In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by . The general hypergeometric function is a function that is (more or less) de ...
s, by Aomoto,
Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
and others; and applications for example to the combinatorics of arranging a number of
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
s in complex ''N''-space (see
arrangement of hyperplanes In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set ''A'' of hyperplanes in a linear, affine, or projective space ''S''. Questions about a hyperplane arrangement ''A'' generally concern geometrical, ...
). Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
s. Their importance and role can be understood through the following example: the hypergeometric series 2''F''1 has the
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
as a special case, and when considered in the form of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...
, 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 3''F''2 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''.


See also

*
Appell series In mathematics, Appell series are a set of four hypergeometric series ''F''1, ''F''2, ''F''3, ''F''4 of two variables that were introduced by and that generalize Gauss's hypergeometric series 2''F''1 of one variable. Appell established the set ...
*
Humbert series In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variable (mathematics), variables that generalize confluent hypergeometric function, Kummer's confluent hypergeometric series 1''F''1 of ...
* Kampé de Fériet function * Lauricella hypergeometric series


Notes


References

* * * * * * * (the first edition has ) * (a reprint of this paper can be found i
''Carl Friedrich Gauss, Werke''
p. 125) * * (part 1 treats hypergeometric functions on Lie groups) * * * * * * * (there is a 2008 paperback with ) *


External links



this book is freely downloadable from the internet. *
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
** ** ** ** {{Series (mathematics) Factorial and binomial topics * Ordinary differential equations Mathematical series