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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in which the ratio of successive
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
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 ne ...
. 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 In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
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 s ...
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,
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
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 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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
:\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 a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s in ''n''. For example, in the case of the series for the exponential function, :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 representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
. 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) . \end ...
:\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 (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...
, 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 ne ...
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 In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...
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 ...
:\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-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is ...
, 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 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 In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero wh ...
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: : (a_i-b_j+1)_pF_q(...a_i..;...,b_j...;z) = a_i\,_pF_q(...a_i+1..;...,b_j...;z) -(b_j-1)_pF_q(...a_i..;...,b_j-1...;z). : (a_i-a_j)_pF_q(...a_i..a_j..;.....;z) = a_i\,_pF_q(...a_i+1..a_j..;......;z) -a_j\,_pF_q(...a_i..a_j+1...;....;z). : b_j\,_pF_q(...a_i....;..b_j...;z) = a_i\,_pF_q(...a_i+1....;..b_j+1...;z) +(b_j-a_i)_pF_q(...a_i....;..b_j+1...;z) . : (a_i-1)_pF_q(...a_i..a_j;...;z) = (a_i-a_j-1)_pF_q(...a_i-1..a_j;...;z) +a_j\,_pF_q(...a_i-1..a_j+1;...;z) . 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 A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
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 .


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. According to , it was in fact first discovered by Pfaff in 1797.


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. 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 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, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
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 s ...
or the
hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
; 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 0''F''1

The functions of the form _0F_1(;a;z) are called confluent hypergeometric limit functions and are closely related to
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
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.) A special case is: :_0F_1\left(;\frac;-\frac\right)=\cos z


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 a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
with ratio ''z'' and coefficient 1. :z ~ _1F_0(2;;z) = \sum_ n z^n = z (1-z)^ is also useful.


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 nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) which is a second-order linear differential equation. Thi ...
. This implies
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
can be expressed in terms of 1''F''1 as well.


The series 1''F''2

Relations to other functions are known for certain parameter combinations only. The function x\; _1F_2\left(\frac;\frac,\frac;-\frac\right) is the antiderivative of the cardinal sine. With modified values of a_1 and b_1, one obtains the antiderivative of \sin(x^\beta)/x^\alpha. The
Lommel function Lommel () is a Municipalities of Belgium, municipality and City status in Belgium, city in the Belgium, Belgian province of Limburg (Belgium), Limburg. Lying in the Campine, Kempen, it has about 34,000 inhabitants and is part of the arrondissement ...
is s_ (z) = \frac _1F_2(1; \frac - \frac + \frac , \frac + \frac + \frac ;-\frac) .


The series 2''F''0

The confluent hypergeometric function of the second kind can be written as: :U(a,b,z) = z^ \; _2 F_0 \left( a, a-b+1; ; -\frac\right).


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 (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
, 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 In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
. 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 * Chris ...
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, classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta ...
. 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 mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
s and
Chebyshev polynomial The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, 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 tr ...
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

The Mott polynomials can be written as: :s_n(x)=(-x/2)^n_3F_0(-n,\frac,1-\frac;;-\frac).


The series 3''F''2

The function ::\operatorname_2(x) = \sum_\, = x \; _3F_2(1,1,1;2,2;x) is the
dilogarithm In mathematics, the dilogarithm (or Spence's function), 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 \tex ...
The function ::Q_n(x;a,b,N)= _3F_2(-n,-x,n+a+b+1;a+1,-N+1;1) is a Hahn polynomial.


The series 4''F''3

The function ::p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \; _4F_3\left( -n, a+b+c+d+n-1, a-t, a+t ; a+b, a+c, a+d ;1\right) is a Wilson polynomial. All roots of a
quintic equation In mathematics, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other word ...
can be expressed in terms of radicals and the Bring radical, which is the real solution to x^5 + x + a = 0. The Bring radical can be written as: ::\operatorname(a) = -a \; _4F_3\left( \frac, \frac, \frac, \frac ; \frac, \frac, \frac ; \frac \right). The partition function Z(K) of the 2D isotropic
Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
with no external magnetic field was found by Onsager in the 1940s and can be expressed as ::\ln Z(K) = \ln(2 \cosh 2K) -k^2 _4F_3\left( 1, 1, \frac, \frac ; 2, 2, 2; 16k^2 \right), with K=\frac and k=\frac\tanh 2K\,\operatorname 2K.


The series q+1''F''q

The functions ::\operatorname_q(z)=z \; _F_q\left(1,1,\ldots,1;2,2,\ldots,2;z\right) ::\operatorname_(z)=z \; _pF_\left(2,2,\ldots,2;1,1,\ldots,1;z\right) for q\in\mathbb_0 and p\in\mathbb are the
Polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
. For each integer ''n''≥2, the roots of the polynomial ''x''''n''−''x''+t can be expressed as a sum of at most ''N''−1 hypergeometric functions of type ''n''+1F''n'', which can always be reduced by eliminating at least one pair of ''a'' and ''b'' parameters.


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 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; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the
q-series In the mathematical field 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 Pochhamme ...
analogues, called the
basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are q-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is ...
, 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 Leg ...
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 In mathematics, an elliptic hypergeometric series is a series Σ''c'n'' such that the ratio ''c'n''/''c'n''−1 is an elliptic function of ''n'', analogous to generalized hypergeometric series where the ratio is a rational function of ...
, are those series where the ratio of terms is an
elliptic function In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
(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 functions, by Aomoto,
Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teache ...
and others; and applications for example to the combinatorics of arranging a number of
hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s in complex ''N''-space (see arrangement of hyperplanes). 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 (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s. Their importance and role can be understood through the following example: the hypergeometric series 2''F''1 has the
Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
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. The table of spherical harmonics co ...
, 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 ...
. 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 (mathematics), series are a set of four hypergeometric series ''F''1, ''F''2, ''F''3, ''F''4 of two variable (mathematics), variables that were introduced by and that generalize hypergeometric function, Gauss's hyper ...
* Humbert series * 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) (a translation is available on Wikisource) * * (part 1 treats hypergeometric functions on Lie groups) * * * * * * * (there is a 2008 paperback with ) *


External links


The book "A = B"
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 Series (mathematics)