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''Gauss series''

A series of the form

$$ 
F ( \alpha , \beta ;  \gamma ;  z) =
$$

$$ 
= \ 
1 + \sum _ {n = 1 } ^  \infty   
\frac{\alpha \dots ( \alpha + n - 1) \beta \dots
( \beta + n - 1) }{n! \gamma \dots ( \gamma + n - 1) }
 z  ^ {n} .
$$

Such a series is meaningful if  $  \gamma $
is not equal to zero or to a negative integer; it converges for  $  | z | < 1 $. 
If, in addition,  $   \mathop{\rm Re} ( \gamma - \alpha - \beta ) > 0 $, 
it also converges for  $  z = 1 $. 
In such a case the Gauss formula

$$ 
F ( \alpha , \beta ;  \gamma ;  1)  = \ 

\frac{\Gamma ( \gamma ) \Gamma ( \gamma - \alpha - \beta ) }{\Gamma ( \gamma - \alpha ) \Gamma ( \gamma - \beta ) }
 ,
$$

where  $  \Gamma ( z) $
is the gamma-function, holds. An analytic function defined with the aid of a hypergeometric series is said to be a [[Hypergeometric function|hypergeometric function]].

A generalized hypergeometric series is a series of the form

$$ 
{} _ {p} F _ {q} ( \alpha _ {1} \dots \alpha _ {p} ; \ 
\gamma _ {1} \dots \gamma _ {q} ;  z) =
$$

$$ 
= \ 
\sum _ {n = 0 } ^  \infty   {
\frac{1}{n!}
 } 
\frac{( \alpha _ {1} ) _ {n} \dots ( \alpha _ {p} ) _ {n} }{( \gamma _ {1} ) _ {n} \dots ( \gamma _ {q} ) _ {n} }
 z  ^ {n} ,
$$

where  $  ( x) _ {n} \equiv x( x + 1) \dots ( x + n - 1) $. 
If this notation is used, the series

is written as  $  {} _ {2} F _ {1} ( \alpha , \beta ;  \gamma ;  z) $.

====Comments====
Generalized hypergeometric series can be characterized as power series  $  \sum _ {n = 0 }   ^  \infty  A _ {n} z  ^ {n} $
such that  $  A _ {n + 1 }  /A _ {n} $
is a rational function of  $  n $. 
An analogous characterization for series in two variables was given by J. Horn. This yields a class of power series in two variables which includes the various Appell's hypergeometric series, cf. [[#References|[a1]]].

Basic hypergeometric series can be characterized as power series  $  \sum _ {n = 0 }   ^  \infty  A _ {n} z  ^ {n} $
such that  $  A _ {n + 1 }  /A _ {n} $
is a rational function of  $  q  ^ {n} $, 
where  $  q $
is a fixed complex number not equal to 0 or 1. Such series have the form

$$ 
{} _ {r} \phi _ {s} ( a _ {1} \dots a _ {r} ; \ 
b _ {1} \dots b _ {s} ; \ 
q, z) =
$$

$$ 
= \ 
\sum _ {n = 0 } ^  \infty   
\frac{[ ( - 1 )  ^ {n} q ^ {n ( n -
1)/2 } ] ^ {s - r + 1 } }{( q;  q ) _ {n} }
 
\frac{(
a _ {1} ;  q ) _ {n} \dots ( a _ {r} ;  q ) _ {n} }{
( b _ {1} ;  q ) _ {n} \dots ( b _ {s} ;  q ) _ {n} }
 z  ^ {n} ,
$$

where  $  ( x;  q) _ {n} \equiv ( 1 - x) ( 1 - qx) \dots ( 1 - q ^ {n - 1 } x) $. 
See [[#References|[a2]]].

Hypergeometric functions of matrix argument have also been studied, cf. [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Appell, M.J. Kampé de Fériet, "Fonctions hypergéométriques et hypersphériques: Polynômes d'Hermite" , Gauthier-Villars  (1926) {{ZBL|52.0361.13}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press  (1989)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Gross, D. Richards, "Special functions of matrix argument I"  ''Trans. Amer. Math. Soc.'' , '''301'''  (1987)  pp. 781–811</TD></TR>
</table>