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{{Short description|Family of power series in mathematics}}
 [[File:Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2iPlot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D]]
In [[Mathematics|mathematics]], a '''generalized hypergeometric series''' is a [[Power series|power series]] in which the ratio of successive [[Coefficient|coefficient]]s indexed by ''n'' is a [[Rational function|rational function]] 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|analytic continuation]]. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include  the (Gaussian) [[Hypergeometric function|hypergeometric function]] and the [[Confluent hypergeometric function|confluent hypergeometric function]] as special cases, which in turn have many particular [[Special functions|special functions]] as special cases, such as elementary functions, [[Bessel function]]s, and the [[Orthogonal polynomials|classical orthogonal polynomials]].

==Notation==
A hypergeometric series is formally defined as a [[Power series|power series]] 
:<math>\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{n \geqslant 0} \beta_n z^n</math>
in which the ratio of successive coefficients is a [[Rational function|rational function]] of ''n''. That is,
:<math>\frac{\beta_{n+1}}{\beta_n} = \frac{A(n)}{B(n)}</math>
where ''A''(''n'') and ''B''(''n'') are [[Polynomial|polynomial]]s in ''n''.

For example, in the case of the series for the [[Exponential function|exponential function]],
:<math>1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots,</math>
we have:
:<math> \beta_n = \frac{1}{n!}, \qquad \frac{\beta_{n+1}}{\beta_n} = \frac{1}{n+1}.</math>
So this satisfies the definition with {{math|''A''(''n'') {{=}} 1}} and {{math|''B''(''n'') {{=}} ''n'' + 1}}.

It is customary to factor out the leading term, so β<sub>0</sub> is assumed to be 1. The polynomials can be factored into linear factors of the form (''a<sub>j</sub>''&nbsp;+&nbsp;''n'') and (''b''<sub>''k''</sub>&nbsp;+&nbsp;''n'') respectively, where the ''a''<sub>''j''</sub> and ''b''<sub>''k''</sub> are complex numbers.

For historical reasons, it is assumed that (1&nbsp;+&nbsp;''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 
:<math>\frac{c(a_1+n)\cdots(a_p+n)}{d(b_1+n)\cdots(b_q+n)(1+n)}</math>,
where ''c'' and ''d'' are the leading coefficients of ''A'' and ''B''. The series then has the form
:<math>1 + \frac{a_1\cdots a_p}{b_1\cdots b_q \cdot 1}\frac{cz}{d} + \frac{a_1\cdots a_p}{b_1\cdots b_q \cdot 1} \frac{(a_1+1)\cdots(a_p+1)}{(b_1+1)\cdots (b_q+1) \cdot 2}\left(\frac{cz}{d}\right)^2+\cdots</math>,
or, by scaling ''z'' by the appropriate factor and rearranging,
:<math>1 + \frac{a_1\cdots a_p}{b_1\cdots b_q}\frac{z}{1!} + \frac{a_1(a_1+1)\cdots a_p(a_p+1)}{b_1(b_1+1)\cdots b_q(b_q+1)}\frac{z^2}{2!}+\cdots</math>.

This has the form of an [[Generating function|exponential generating function]]. This series is usually denoted by
:<math>{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)</math>

or

:<math>\,{}_pF_q \left[\begin{matrix}  a_1 & a_2 & \cdots & a_{p} \\ b_1 & b_2 & \cdots & b_q \end{matrix} ; z \right].</math>

Using the rising factorial or Pochhammer symbol

:<math>\begin{align}
(a)_0 &= 1, \\
(a)_n &= a(a+1)(a+2) \cdots (a+n-1), && n \geq 1
\end{align}</math>

this can be written

:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!}.</math>

(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|radius of convergence]], then the series defines an [[Analytic function|analytic function]]. Such a function, and its [[Analytic continuation|analytic continuation]]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|incomplete gamma function]] has the [[Asymptotic expansion|asymptotic expansion]]
:<math>\Gamma(a,z) \sim z^{a-1}e^{-z}\left(1+\frac{a-1}{z}+\frac{(a-1)(a-2)}{z^2} + \cdots\right)</math>
which could be written ''z''<sup>''a''−1</sup>''e''<sup>−z</sup>&nbsp;<sub>2</sub>''F''<sub>0</sub>(1−''a'',1;;−''z''<sup>−1</sup>). 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|basic hypergeometric series]], which, despite its name, is a rather more complicated and recondite series.  The "basic" series is the [[Q-analog|q-analog]] of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from [[Zonal spherical function|zonal spherical function]]s on [[Symmetric space|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|bilateral hypergeometric series]].

==Convergence conditions==
There are certain values of the ''a''<sub>''j''</sub> and ''b''<sub>''k''</sub> for which the numerator or the denominator of the coefficients is 0.
* If any ''a''<sub>''j''</sub> 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''<sub>''j''</sub>.
* If any ''b''<sub>''k''</sub> is a non-positive integer (excepting the previous case with ''b''<sub>''k''</sub> &lt; ''a''<sub>''j''</sub>) then the denominators become 0 and the series is undefined.

Excluding these cases, the [[Ratio test|ratio test]] can be applied to determine the radius of convergence.
* If ''p'' &lt; ''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''|&nbsp;&lt;&nbsp;1 and diverges for |''z''|&nbsp;&gt;&nbsp;1. Whether it converges for |''z''|&nbsp;=&nbsp;1 is more difficult to determine. Analytic continuation can be employed for larger values of ''z''.
* If ''p'' &gt; ''q'' + 1 then the ratio of coefficients grows without bound. This implies that, besides ''z''&nbsp;=&nbsp;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 
:<math>\Re\left(\sum b_k - \sum a_j\right)>0</math>.
Further, if ''p''=''q''+1, <math>\sum_{i=1}^{p}a_{i}\geq\sum_{j=1}^{q}b_{j}</math> and ''z'' is real, then the following convergence result holds {{harvtxt|Quigley|Wilson|Walls|Bedford|2013}}:
:<math>\lim_{z\rightarrow 1}(1-z)\frac{d\log(_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z^{p}))}{dz}=\sum_{i=1}^{p}a_{i}-\sum_{j=1}^{q}b_{j}</math>.

==Basic properties==
It is immediate from the definition that the order of the parameters ''a<sub>j</sub>'', or the order of the parameters ''b<sub>k</sub>'' can be changed without changing the value of the function. Also, if any of the parameters ''a<sub>j</sub>'' is equal to any of the parameters ''b<sub>k</sub>'', then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. For example,
:<math>\,{}_2F_1(3,1;1;z) = \,{}_2F_1(1,3;1;z) = \,{}_1F_0(3;;z)</math>.
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.<ref>{{cite book|first=A. P.|last1=Prudnikov|first2=Yu. A.|last2=Brychkov|first3=O. I.|last3=Marichev|title=Integrals & Series Volume 3: More Special Functions|publisher=Gordon and Breach|year=1990|pages=439}}</ref>
:<math>
{}_{A+1}F_{B+1}\left[ 
\begin{array}{c}
a_{1},\ldots ,a_{A},c + n \\ 
b_{1},\ldots ,b_{B},c
\end{array}
;z\right] = \sum_{j = 0}^n \binom{n}{j} \frac{z^j}{(c)_j} \frac{\prod_{i = 1}^A (a_i)_j}{\prod_{i = 1}^B (b_i)_j} {}_AF_B\left[ 
\begin{array}{c}
a_{1} + j,\ldots ,a_{A} + j \\ 
b_{1} + j,\ldots ,b_{B} + j
\end{array}
;z\right]
</math>

===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<ref>{{harv|Slater|1966|loc=Equation (4.1.2)}}</ref>

:<math> {}_{A+1}F_{B+1}\left[ 
\begin{array}{c}
a_{1},\ldots ,a_{A},c \\ 
b_{1},\ldots ,b_{B},d
\end{array}
;z\right] =\frac{\Gamma (d)}{\Gamma (c)\Gamma (d-c)}
\int_{0}^{1}t^{c-1}(1-t)_{{}}^{d-c-1}\ {}_{A}F_{B}\left[ 
\begin{array}{c}
a_{1},\ldots ,a_{A} \\ 
b_{1},\ldots ,b_{B}
\end{array} ; tz\right]  dt</math>

===Differentiation===
The generalized hypergeometric function satisfies
:<math>\begin{align}
\left (z\frac{{\rm{d}}}{{\rm{d}}z} + a_j \right ){}_pF_q\left[ \begin{array}{c} a_1,\dots,a_j,\dots,a_p \\ b_1,\dots,b_q\end{array} ;z\right] &= a_j \; {}_pF_q\left[ \begin{array}{c} a_1,\dots,a_j+1,\dots,a_p \\ b_1,\dots,b_q \end{array} ;z\right] \\
\end{align}</math>

and  

<math>\begin{align}
\left (z\frac{{\rm{d}}}{{\rm{d}}z} + b_k - 1 \right ){}_pF_q\left[ \begin{array}{c} a_1,\dots,a_p \\ b_1,\dots,b_k,\dots,b_q\end{array} ;z\right] &= (b_k - 1) \; {}_pF_q\left[ \begin{array}{c} a_1,\dots,a_p \\ b_1,\dots,b_k-1,\dots,b_q \end{array} ;z \right] \text{ for } b_k\neq 1
\end{align}</math>

Additionally,

<math>\begin{align}
\frac{{\rm{d}}}{{\rm{d}}z} \; {}_pF_q\left[ \begin{array}{c} a_1,\dots,a_p \\ b_1,\dots,b_q \end{array} ;z \right] &= \frac{\prod_{i=1}^p a_i}{\prod_{j=1}^q b_j}\; {}_pF_q\left[ \begin{array}{c} a_1+1,\dots,a_p+1 \\ b_1+1,\dots,b_q+1 \end{array} ;z \right]
\end{align}</math>

Combining these gives a differential equation satisfied by ''w'' = <sub>''p''</sub>''F''<sub>''q''</sub>:
:<math>z\prod_{n=1}^{p}\left(z\frac{{\rm{d}}}{{\rm{d}}z} + a_n\right)w = z\frac{{\rm{d}}}{{\rm{d}}z}\prod_{n=1}^{q}\left(z\frac{{\rm{d}}}{{\rm{d}}z} + b_n-1\right)w</math>.

==Contiguous function and related identities==
Take the following operator:
:<math>\vartheta = z\frac{{\rm{d}}}{{\rm{d}}z}.</math> 
From the differentiation formulas given above, the linear space spanned by 
:<math>{}_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)</math> 
contains each of 
:<math>{}_pF_q(a_1,\dots,a_j+1,\dots,a_p;b_1,\dots,b_q;z),</math> 
:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_k-1,\dots,b_q;z),</math>
:<math>z\; {}_pF_q(a_1+1,\dots,a_p+1;b_1+1,\dots,b_q+1;z),</math>
:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z).</math>
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 <math>{}_pF_q</math>.

For example, in the simplest non-trivial case,
:<math> \; {}_0F_1(;a;z) = (1) \; {}_0F_1(;a;z)</math>,
:<math> \; {}_0F_1(;a-1;z) = (\frac{\vartheta}{a-1}+1) \; {}_0F_1(;a;z)</math>,
:<math>z \; {}_0F_1(;a+1;z) = (a\vartheta) \; {}_0F_1(;a;z)</math>,
So 
:<math> \; {}_0F_1(;a-1;z)- \; {}_0F_1(;a;z) = \frac{z}{a(a-1)} \; {}_0F_1(;a+1;z)</math>.

This, and other important examples,

:<math> \; {}_1F_1(a+1;b;z)- \, {}_1F_1(a;b;z) = \frac{z}{b} \; {}_1F_1(a+1;b+1;z)</math>,
:<math> \; {}_1F_1(a;b-1;z)- \, {}_1F_1(a;b;z) = \frac{az}{b(b-1)} \; {}_1F_1(a+1;b+1;z)</math>,
:<math> \; {}_1F_1(a;b-1;z)- \, {}_1F_1(a+1;b;z) = \frac{(a-b+1)z}{b(b-1)} \; {}_1F_1(a+1;b+1;z)</math>
:<math> \; {}_2F_1(a+1,b;c;z)- \, {}_2F_1(a,b;c;z) = \frac{bz}{c} \; {}_2F_1(a+1,b+1;c+1;z)</math>,
:<math> \; {}_2F_1(a+1,b;c;z)- \, {}_2F_1(a,b+1;c;z) = \frac{(b-a)z}{c} \; {}_2F_1(a+1,b+1;c+1;z)</math>,
:<math> \; {}_2F_1(a,b;c-1;z)- \, {}_2F_1(a+1,b;c;z) = \frac{(a-c+1)bz}{c(c-1)} \; {}_2F_1(a+1,b+1;c+1;z)</math>,

can be used to generate [[Continued fraction|continued fraction]] expressions known as [[Gauss's continued fraction]].

Similarly, by applying the differentiation formulas twice, there are <math>\binom{p+q+3}{2}</math> such functions contained in 
:<math>\{1, \vartheta, \vartheta^2\}\; {}_p F_q (a_1,\dots,a_p;b_1,\dots,b_q;z),</math> 
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''<sub>''j''</sub>, ''b''<sub>''k''</sub> in 
:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)</math> 
is called '''contiguous''' to 
:<math>{}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z).</math>
Using the technique outlined above, an identity relating <math>{}_0F_1(;a;z)</math> and its two contiguous functions can be given, six identities relating <math>{}_1F_1(a;b;z)</math> and any two of its four contiguous functions, and fifteen identities relating <math>{}_2F_1(a,b;c;z)</math> 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<ref>See {{harv|Slater|1966|loc=Section&nbsp;2.3.1}} or {{harv|Bailey|1935|loc=Section&nbsp;2.2}} for a proof.</ref> {{harv|Saalschütz|1890}} is
:<math>{}_3F_2 (a,b, -n;c, 1+a+b-c-n;1)= \frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}.</math>
For extension of this theorem, see a research paper by Rakha & Rathie.

===Dixon's identity===
{{Main|Dixon's identity}}

Dixon's identity,<ref>See {{harv|Bailey|1935|loc=Section&nbsp;3.1}} for a detailed proof. An alternative proof is in {{harv|Slater|1966|loc=Section&nbsp;2.3.3}}</ref> first proved by {{harvtxt|Dixon|1902}}, gives the sum of a well-poised <sub>3</sub>''F''<sub>2</sub> at 1:
:<math>{}_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac{\Gamma(1+\frac{a}{2})\Gamma(1+\frac{a}{2}-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)}.</math>
For generalization of Dixon's identity, see a paper by Lavoie, et al.

===Dougall's formula===
Dougall's formula {{harvs|first=|last=Dougall|year=1907}} gives the sum of a very well-poised series that is terminating and 2-balanced.
:<math>\begin{align}
{}_7F_6 & \left(\begin{matrix}a&1+\frac{a}{2}&b&c&d&e&-m\\&\frac{a}{2}&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\ \end{matrix};1\right) = \\
&=\frac{(1+a)_m(1+a-b-c)_m(1+a-c-d)_m(1+a-b-d)_m}{(1+a-b)_m(1+a-c)_m(1+a-d)_m(1+a-b-c-d)_m}.
\end{align}</math>

Terminating means that ''m'' is a non-negative integer and 2-balanced means that
:<math>1+2a=b+c+d+e-m.</math>
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 <sub>2</sub>''F''<sub>2</sub>===
'''Identity 1.'''
:<math>e^{-x} \; {}_2F_2(a,1+d;c,d;x)= {}_2F_2(c-a-1,f+1;c,f;-x)</math>
where 
:<math>f=\frac{d(a-c+1)}{a-d}</math>;

'''Identity 2.'''
:<math>e^{-\frac x 2} \, {}_2F_2 \left(a, 1+b; 2a+1, b; x\right)= {}_0F_1 \left(;a+\tfrac{1}{2}; \tfrac {x^2} {16}\right) - \frac{x\left(1-\tfrac{2a} b\right)}{2(2a+1)}\; {}_0F_1 \left(;a+\tfrac 3 2; \tfrac {x^2} {16}\right),</math>
which links [[Bessel function]]s to <sub>2</sub>''F''<sub>2</sub>; this reduces to Kummer's second formula for ''b'' = 2''a'':

'''Identity 3.'''
:<math>e^{-\frac x 2} \, {}_1F_1(a,2a,x)= {}_0F_1 \left (;a+\tfrac 1 2; \tfrac{x^2}{16} \right )</math>.

'''Identity 4.'''
:<math>\begin{align}
{}_2F_2(a,b;c,d;x)=& \sum_{i=0} \frac{{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}} \; {}_1F_1(a+i;c+i;x)\frac{x^i}{i!} \\
=& e^x \sum_{i=0} \frac{{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}} \; {}_1F_1(c-a;c+i;-x)\frac{x^i}{i!},
\end{align}</math>
which is a finite sum if ''b-d'' is a non-negative integer.

===Kummer's relation===
Kummer's relation is
:<math>{}_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).</math>

===Clausen's formula===
{{Main|Clausen's formula}}
Clausen's formula
:<math>{}_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</math>
was used by [[Biography:Louis de Branges de Bourcia|de Branges]] to prove the Bieberbach conjecture.

==Special cases==
Many of the special functions in mathematics are special cases of the [[Confluent hypergeometric function|confluent hypergeometric function]] or the [[Hypergeometric function|hypergeometric function]]; see the corresponding articles for examples.

===The series <sub>0</sub>''F''<sub>0</sub>===
{{Main|Exponential function}}
As noted earlier, <math>{}_0F_0(;;z) = e^z</math>. The differential equation for this function is <math>\frac{d}{dz}w = w</math>, which has solutions <math>w = ke^z</math> where ''k'' is a constant.

===The series <sub>0</sub>''F''<sub>1</sub>===
The functions of the form <math>{}_0F_1(;a;z)</math> are called '''confluent hypergeometric limit functions''' and are closely related to [[Bessel function]]s. 

The relationship is: 
:<math>J_\alpha(x)=\frac{(\tfrac{x}{2})^\alpha}{\Gamma(\alpha+1)}  {}_0F_1\left  (;\alpha+1; -\tfrac{1}{4}x^2 \right ).</math> 
:<math>I_\alpha(x)=\frac{(\tfrac{x}{2})^\alpha}{\Gamma(\alpha+1)}  {}_0F_1\left  (;\alpha+1; \tfrac{1}{4}x^2 \right ).</math> 
The differential equation for this function is 
:<math>w = \left (z\frac{d}{dz}+a \right )\frac{dw}{dz}</math> 
or 
:<math>z\frac{d^2w}{dz^2}+a\frac{dw}{dz}-w = 0.</math> 
When ''a'' is not a positive integer, the substitution 
:<math>w = z^{1-a}u,</math> 
gives a linearly independent solution 
:<math>z^{1-a}\;{}_0F_1(;2-a;z),</math> 
so the general solution is 
:<math>k\;{}_0F_1(;a;z)+l z^{1-a}\;{}_0F_1(;2-a;z)</math> 
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:
:<math>{}_0F_1\left(;\frac{1}{2};-\frac{z^2}{4}\right)=\cos z</math>

===The series <sub>1</sub>''F''<sub>0</sub>===
{{Main|Binomial series}}
An important case is: 
:<math>{}_1F_0(a;;z) = (1-z)^{-a}.</math> 
The differential equation for this function is 
:<math>\frac{d}{dz}w =\left  (z\frac{d}{dz}+a \right )w,</math> 
or 
:<math>(1-z)\frac{dw}{dz} = aw,</math> 
which has solutions 
:<math>w=k(1-z)^{-a}</math> 
where ''k'' is a constant.

:<math>{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}</math> is the [[Geometric series|geometric series]] with ratio ''z'' and coefficient 1.

:<math>z ~ {}_1F_0(2;;z) = \sum_{n \geqslant 0} n z^n = z (1-z)^{-2}</math> is also useful.

===The series <sub>1</sub>''F''<sub>1</sub>===
{{main|Confluent hypergeometric function}}
The functions of the form <math>{}_1F_1(a;b;z)</math> are called '''confluent hypergeometric functions of the first kind''', also written <math>M(a;b;z)</math>. The incomplete gamma function <math>\gamma(a,z)</math> is a special case.

The differential equation for this function is

:<math>\left (z\frac{d}{dz}+a \right )w = \left (z\frac{d}{dz}+b \right )\frac{dw}{dz}</math>

or

:<math>z\frac{d^2w}{dz^2}+(b-z)\frac{dw}{dz}-aw = 0.</math>

When ''b'' is not a positive integer, the substitution

:<math>w = z^{1-b}u,</math>

gives a linearly independent solution

:<math>z^{1-b}\;{}_1F_1(1+a-b;2-b;z),</math>

so the general solution is

:<math>k\;{}_1F_1(a;b;z)+l z^{1-b}\;{}_1F_1(1+a-b;2-b;z)</math>

where ''k'', ''l'' are constants.

When a is a non-positive integer, −''n'', <math>{}_1F_1(-n;b;z)</math> is a polynomial. Up to constant factors, these are the [[Laguerre polynomials]]. This implies [[Hermite polynomials]] can be expressed in terms of <sub>1</sub>''F''<sub>1</sub> as well.

===The series <sub>1</sub>''F''<sub>2</sub>===

Relations to other functions are known for certain parameter combinations only.

The function <math>x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right)</math> is the antiderivative of the cardinal sine. With modified values of <math>a_1</math> and <math>b_1</math>, one obtains the antiderivative of <math>\sin(x^\beta)/x^\alpha</math>.<ref>Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)</ref>

It has been proposed that <math>{}_1F_2\left(n-\frac{1}{2};n+1,2n+1;-{x^2}\right)</math> can be expressed through the Bessel function <math>J_n(x)</math> and its derivative.<ref>https://math.stackexchange.com/questions/3978473 (2021)</ref>

The function <math>{}_1F_2(1;a,a+1;x)</math> is essentially a [[Lommel function]].<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10), according to https://mathoverflow.net/questions/98684</ref>

===The series <sub>2</sub>''F''<sub>0</sub>===
The confluent hypergeometric function of the second kind can be written as:<ref>{{cite web |title=DLMF: §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions |url=https://dlmf.nist.gov/13.6 |website=dlmf.nist.gov}}</ref>
:<math>U(a,b,z) = z^{-a} \; {}_2 F_0 \left( a, a-b+1; ; -\frac{1}{z}\right).</math>

===The series <sub>2</sub>''F''<sub>1</sub>===
{{main|Hypergeometric function}}
Historically, the most important are the functions of the form <math>{}_2F_1(a,b;c;z)</math>. 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 <sub>''p''</sub>''F''<sub>''q''</sub> if there is risk of confusion. This function was first studied in detail by [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]], who explored the conditions for its convergence.

The differential equation for this function is

:<math> \left (z\frac{d}{dz}+a \right ) \left (z\frac{d}{dz}+b \right )w =\left  (z\frac{d}{dz}+c \right )\frac{dw}{dz}</math>

or

:<math>z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0.</math>

It is known as the hypergeometric differential equation. When ''c'' is not a positive integer, the substitution

:<math>w = z^{1-c}u</math>

gives a linearly independent solution

:<math> z^{1-c}\; {}_2F_1(1+a-c,1+b-c;2-c;z),</math>

so the general solution for |''z''| < 1 is

:<math>k\; {}_2F_1(a,b;c;z)+l z^{1-c}\; {}_2F_1(1+a-c,1+b-c;2-c;z)</math>

where ''k'', ''l'' are constants. Different solutions can be derived for other values of ''z''. In fact there are 24 solutions, known as the [[Biography:Ernst Kummer|Kummer]] solutions, derivable using various identities, valid in different regions of the complex plane.

When ''a'' is a non-positive integer, −''n'',
 
:<math>{}_2F_1(-n,b;c;z)</math>

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 <sub>2</sub>''F''<sub>1</sub> 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.:

:<math>\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.</math>

===The series <sub>3</sub>''F''<sub>0</sub>===
This occurs in connection with [[Mott polynomials]].<ref>See Erdélyi et al. 1955.</ref>

===The series <sub>3</sub>''F''<sub>2</sub>===

The function
::<math>\operatorname{Li}_2(x) = \sum_{n>0}\,{x^n}{n^{-2}} = x \; {}_3F_2(1,1,1;2,2;x)</math>
is the [[Dilogarithm|dilogarithm]]<ref>{{cite web|last=Candan|first=Cagatay|title=A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x
|url=http://www.eee.metu.edu.tr/~ccandan/pub_dir/hyper_rel.pdf}}</ref>

The function
::<math>Q_n(x;a,b,N)= {}_3F_2(-n,-x,n+a+b+1;a+1,-N+1;1)</math>
is a Hahn polynomial.

===The series <sub>4</sub>''F''<sub>3</sub>===

The function
::<math>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)</math>
is a Wilson polynomial.

All roots of a quintic equation can be expressed in terms of radicals and the [[Bring radical]], which is the real solution to <math>x^5 + x + a = 0</math>. The Bring radical can be written as:<ref name="BR">
{{cite arXiv
 | last = Glasser | first = M. Lawrence
 | year = 1994
 | title = The quadratic formula made hard: A less radical approach to solving equations
 | eprint = math.CA/9411224
}}
</ref>
::<math>\operatorname{BR}(t) = -a \; {}_4F_3\left( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ; \frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; \frac{3125a^4}{256} \right).</math>

===The series <sub>q+1</sub>''F''<sub>q</sub>===

The functions
::<math>\operatorname{Li}_q(z)=z \; {}_{q+1}F_q\left(1,1,\ldots,1;2,2,\ldots,2;z\right)</math>
::<math>\operatorname{Li}_{-p}(z)=z \; {}_pF_{p-1}\left(2,2,\ldots,2;1,1,\ldots,1;z\right)</math>
for <math>q\in\mathbb{N}_0</math> and <math>p\in\mathbb{N}</math> are the [[Polylogarithm]].

For each integer ''n''≥2, the roots of the polynomial ''x''<sup>''n''</sup>−''x''+t can be expressed as a sum of at most ''N''−1 hypergeometric functions of type <sub>''n''+1</sub>F<sub>''n''</sub>, which can always be reduced by eliminating at least one pair of ''a'' and ''b'' parameters.<ref name="BR"/>

==Generalizations==
The generalized hypergeometric function is linked to the [[Meijer G-function]] and the MacRobert E-function.  Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and Joseph Kampé de Fériet; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the [[Basic hypergeometric series|basic hypergeometric series]], were given by [[Biography:Eduard Heine|Eduard Heine]] in the late nineteenth century. Here, the ratios considered of successive terms, instead of a rational function of ''n'', are a rational function of ''q<sup>n</sup>''. Another generalization, the [[Elliptic hypergeometric series|elliptic hypergeometric series]], are those series where the ratio of terms is an [[Elliptic function|elliptic function]] (a doubly periodic [[Meromorphic function|meromorphic function]]) of ''n''.

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of [[General hypergeometric function|general hypergeometric function]]s, by Aomoto, [[Biography:Israel Gelfand|Israel Gelfand]] and others; and applications for example to the combinatorics of arranging a number of [[Hyperplane|hyperplane]]s in complex ''N''-space (see [[Arrangement of hyperplanes|arrangement of hyperplanes]]).

Special hypergeometric functions occur as [[Zonal spherical function|zonal spherical function]]s on [[Symmetric space|Riemannian symmetric spaces]] and semi-simple [[Lie group]]s. Their importance and role can be understood through the following example: the hypergeometric series <sub>2</sub>''F''<sub>1</sub> has the [[Legendre polynomials]] as a special case, and when considered in the form of [[Physics:Spherical harmonics|spherical harmonics]], these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group [[SO(3)]].  In tensor product decompositions of concrete representations of this group [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]] are met, which can be written as <sub>3</sub>''F''<sub>2</sub> hypergeometric series.

[[Bilateral hypergeometric series]] are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.

[[Fox–Wright function]]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]]
* [[Lauricella hypergeometric series]]

==Notes==
{{Reflist|2}}

==References==
* {{dlmf|id=16|first1=R. A.|last1=Askey|first2=Adri B. Olde|last2= Daalhuis}}
* {{cite book | last1= Andrews | first1= George E. | last2= Askey | first2= Richard | name-list-style= amp | last3= Roy | first3= Ranjan | title= Special functions | publisher= Cambridge University Press | year= 1999 | series= Encyclopedia of Mathematics and its Applications | volume= 71 <!--| isbn= 978-0-521-62321-6 -->|isbn=978-0-521-78988-2 | mr= 1688958 }}
* {{cite book | last= Bailey | first= W.N. | title= Generalized Hypergeometric Series | publisher= Cambridge University Press | location= London | year= 1935 | series= Cambridge Tracts in Mathematics and Mathematical Physics | volume= 32 | zbl= 0011.02303 }}
* {{cite journal | last= Dixon | first= A.C. | title= Summation of a certain series | journal= Proc. London Math. Soc. | year= 1902 | volume= 35 | issue= 1 | pages= 284–291 | doi=10.1112/plms/s1-35.1.284 | jfm= 34.0490.02 | url= https://zenodo.org/record/1433433 }}
* {{cite journal | last= Dougall | first= J. | title= On Vandermonde's theorem and some more general expansions | journal= Proc. Edinburgh Math. Soc. | year= 1907 | volume= 25 | pages= 114–132 | doi= 10.1017/S0013091500033642 | doi-access= free }}
*{{Cite book | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm |  last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol. III | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | mr=0066496 | year=1955}}
* {{cite book | last1= Gasper | first1= George | last2= Rahman | first2= Mizan |  title= Basic Hypergeometric Series | edition= 2nd | year= 2004 | series= Encyclopedia of Mathematics and Its Applications | volume= 96 | publisher= Cambridge University Press | location= Cambridge, UK | mr= 2128719 | zbl= 1129.33005 | isbn= 978-0-521-83357-8 }} (the first edition has {{isbn|0-521-35049-2}})
* {{cite journal | last= Gauss | first= Carl Friedrich |  title= Disquisitiones generales circa seriam infinitam &nbsp; <math> 1 + \tfrac {\alpha \beta} {1 \cdot \gamma} ~x + \tfrac {\alpha (\alpha+1) \beta (\beta+1)} {1 \cdot 2 \cdot \gamma (\gamma+1)} ~x~x + \mbox{etc.} </math> | language= la | url= https://books.google.com/books?id=uDMAAAAAQAAJ | journal= Commentationes Societatis Regiae Scientarum Gottingensis Recentiores | location= Göttingen | year= 1813 | volume= 2 }} (a reprint of this paper can be found in [https://archive.org/details/bub_gb_uDMAAAAAQAAJ ''Carl Friedrich Gauss, Werke''], p.&nbsp;125)
*{{Citation | last1=Grinshpan | first1=A. Z. | title=Generalized hypergeometric functions: product identities and weighted norm inequalities | doi=10.1007/s11139-013-9487-x | year=2013 | 
journal=The Ramanujan Journal | volume=31 | issue=1–2 | pages=53–66 | s2cid=121054930 }}

* {{cite book | last1= Heckman | first1= Gerrit | name-list-style= amp | last2= Schlichtkrull | first2= Henrik | title= Harmonic Analysis and Special Functions on Symmetric Spaces | publisher= Academic Press | location= San Diego | year= 1994 | isbn= 978-0-12-336170-7 }} (part 1 treats hypergeometric functions on Lie groups)
* {{cite journal|last1=Lavoie |first1=J.L.|last2=Grondin |first2=F. |last3=Rathie |first3=A.K.
|last4=Arora|first4=K. |title=Generalizations of Dixon's theorem on the sum of a 3F2
|journal=Math. Comp. |volume=62 |issue=205|pages=267–276|year=1994|doi=10.2307/2153407 |jstor=2153407}}
* {{cite journal|first1=A. R. | last1=Miller| first2=R. B. | last2=Paris
|title=Euler-type transformations for the generalized hypergeometric function <sub>r+2</sub>''F''<sub>r+1</sub>
|journal=Z. Angew. Math. Phys. | volume=62|year=2011 | issue=1| pages=31–45 |doi=10.1007/s00033-010-0085-0
| bibcode=2011ZaMP...62...31M| s2cid=30484300| url=https://rke.abertay.ac.uk/en/publications/30e4ad50-271e-40a7-bfb3-dc6515871b50}}
* {{cite journal|last1=Quigley |first1=J.|last2=Wilson |first2=K.J. |last3=Walls |first3=L.
|last4=Bedford|first4=T. |title=A Bayes linear Bayes Method for Estimation of Correlated Event Rates
|journal=Risk Analysis |volume=33|issue=12|pages=2209–2224|year=2013|doi=10.1111/risa.12035|pmid=23551053|s2cid=24476762 |url=https://strathprints.strath.ac.uk/43403/1/EventRatesRA.pdf}}
* {{cite journal | last1= Rathie | first1= Arjun K. | last2= Pogány | first2= Tibor K. | title= New summation formula for <sub>3</sub>''F''<sub>2</sub>(1/2) and a Kummer-type II transformation of <sub>2</sub>''F''<sub>2</sub>(''x'') | journal = Mathematical Communications | volume= 13 | year= 2008 | pages= 63–66 | url= http://hrcak.srce.hr/file/37118 | mr= 2422088 | zbl= 1146.33002 }}
* {{cite journal|last1=Rakha |first1=M.A.|last2=Rathie |first2=Arjun K.|title=Extensions of Euler's type- II transformation and Saalschutz's theorem |journal=Bull. Korean Math. Soc.|volume=48 |number=1|pages=151–156|year=2011 |doi=10.4134/bkms.2011.48.1.151|doi-access=free}}
* {{cite journal | last= Saalschütz | first= L. | title=Eine Summationsformel | language = de | journal= Zeitschrift für Mathematik und Physik | year= 1890 | volume= 35 | pages= 186–188 | jfm=22.0262.03 }}
* {{cite book | last= Slater | first= Lucy Joan |  title= Generalized Hypergeometric Functions | publisher= Cambridge University Press | location= Cambridge, UK | year= 1966 | isbn= 978-0-521-06483-5 | mr= 0201688 | zbl= 0135.28101 }} (there is a 2008 paperback with {{isbn|978-0-521-09061-2}})
* {{cite book | last= Yoshida | first= Masaaki | title= Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces | publisher= Friedr. Vieweg & Sohn | location= Braunschweig/Wiesbaden | year= 1997 | isbn= 978-3-528-06925-4 | mr= 1453580 }}

==External links==
* [https://web.archive.org/web/20060129095451/http://www.cis.upenn.edu/~wilf/AeqB.html The book "A = B"], this book is freely downloadable from the internet.
* [[MathWorld]]
** {{MathWorld |title=Generalized Hypergeometric Function |urlname= GeneralizedHypergeometricFunction}}
** {{MathWorld |title=Hypergeometric Function |urlname= HypergeometricFunction}}
** {{MathWorld |title=Confluent Hypergeometric Function of the First Kind |urlname= ConfluentHypergeometricFunctionoftheFirstKind}}
** {{MathWorld |title=Confluent Hypergeometric Limit Function |urlname= ConfluentHypergeometricLimitFunction}}

{{Series (mathematics)}}
[[Category:Factorial and binomial topics]]
[[Category:Hypergeometric functions|*]]
[[Category:Ordinary differential equations]]
[[Category:Mathematical series]]

{{Sourceattribution|Generalized hypergeometric function}}