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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 const ...
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 var ...
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 rat ...
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 new ...
. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the
Gaussian hypergeometric series Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. 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 by ...
as special cases, such as
elementary functions In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
,
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 In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomi ...
.


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 const ...
:\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 rat ...
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 exa ...
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, a ...
, :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 form a ...
. 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 series ...
. 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 In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
:\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 In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
, 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 an ...
. 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 new ...
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 In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
:\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 In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...
, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the
q-analog In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q' ...
of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from
zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vect ...
s 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 In mathematics, a bilateral hypergeometric series is a series Σ''a'n'' summed over ''all'' integers ''n'', and such that the ratio :''a'n''/''a'n''+1 of two terms is a rational function of ''n''. The definition of the generalized hyp ...
.


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 In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
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 integer ...
expressions known as
Gauss's continued fraction In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several i ...
. 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 The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers. ...
.


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 ...
.


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 succ ...
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 In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions on ...
. 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 In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of  ...
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 refer ...
, 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 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 In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...
. 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 polynomial 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 ...
s and
Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
s. 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.


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 In mathematics, Spence's function, or dilogarithm, denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: :\operatorname_2(z) = -\int_0^z\, du \textz ...


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 In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spec ...
.


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.


Generalizations

The generalized hypergeometric function is linked to the
Meijer G-function In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the ...
and the MacRobert E-function. Hypergeometric series were generalised to several variables, for example by
Paul Emile Appell Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity * Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...
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 mé ...
; 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 sym ...
analogues, called the
basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...
, 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 Legen ...
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. The ...
) 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 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 hyper ...
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 function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vect ...
s 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 additio ...
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 applica ...
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 mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
. In tensor product decompositions of concrete representations of this group
Clebsch–Gordan coefficients In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
are met, which can be written as 3''F''2 hypergeometric series.
Bilateral hypergeometric series In mathematics, a bilateral hypergeometric series is a series Σ''a'n'' summed over ''all'' integers ''n'', and such that the ratio :''a'n''/''a'n''+1 of two terms is a rational function of ''n''. The definition of the generalized hyp ...
are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.
Fox–Wright function In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function ''p'F'q''(''z'') based on ideas of and : _p\ ...
s 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 * Humbert series *
Kampé de Fériet function In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet Marie-Joseph Kampé de Fériet (Paris, 14 May 1893 – Villeneuve d'Ascq, 6 April ...
*
Lauricella hypergeometric series In 1893 Giuseppe Lauricella defined and studied four hypergeometric series ''F'A'', ''F'B'', ''F'C'', ''F'D'' of three variables. They are : : F_A^(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^ for , ''x''1, ...


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 Dig ...
** ** ** ** {{Series (mathematics) Factorial and binomial topics * Ordinary differential equations Mathematical series