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''Gauss series''
 
''Gauss series''
  
 
A series of the form
 
A series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484601.png" /></td> </tr></table>
+
$$
 +
F ( \alpha , \beta ; \gamma ; z) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484602.png" /></td> </tr></table>
+
$$
 +
= \
 +
1 + \sum _ {n = 1 } ^  \infty 
 +
\frac{\alpha \dots ( \alpha + n - 1) \beta \dots
 +
( \beta + n - 1) }{n! \gamma \dots ( \gamma + n - 1) }
 +
z  ^ {n} .
 +
$$
  
Such a series is meaningful if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484603.png" /> is not equal to zero or to a negative integer; it converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484604.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484605.png" />, it also converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484606.png" />. In such a case the Gauss formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484607.png" /></td> </tr></table>
+
$$
 +
F ( \alpha , \beta ; \gamma ; 1)  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484608.png" /> 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]].
+
\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
 
A generalized hypergeometric series is a series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484609.png" /></td> </tr></table>
+
$$
 
+
{} _ {p} F _ {q} ( \alpha _ {1} \dots \alpha _ {p} ; \
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846010.png" /></td> </tr></table>
+
\gamma _ {1} \dots \gamma _ {q} ; z) =
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846011.png" />. If this notation is used, the series
 
  
is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846012.png" />.
+
$$
 +
= \
 +
\sum _ {n = 0 } ^  \infty  {
 +
\frac{1}{n!}
 +
}
 +
\frac{( \alpha _ {1} ) _ {n} \dots ( \alpha _ {p} ) _ {n} }{( \gamma _ {1} ) _ {n} \dots ( \gamma _ {q} ) _ {n} }
 +
z  ^ {n} ,
 +
$$
  
 +
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====
 
====Comments====
Generalized hypergeometric series can be characterized as power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846014.png" /> is a rational function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846015.png" />. 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846017.png" /> is a rational function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846019.png" /> is a fixed complex number not equal to 0 or 1. Such series have the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846020.png" /></td> </tr></table>
+
$$
 +
{} _ {r} \phi _ {s} ( a _ {1} \dots a _ {r} ; \
 +
b _ {1} \dots b _ {s} ; \
 +
q, z) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846021.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {n = 0 } ^  \infty 
 +
\frac{[ ( - 1 )  ^ {n} q ^ {n ( n -
 +
1)/2 } ] ^ {s - r + 1 } }{( q; q ) _ {n} }
 +
 +
\frac{(
 +
a _ {1} ;  q ) _ {n} \dots ( a _ {r} ;  q ) _ {n} }{
 +
( b _ {1} ; q ) _ {n} \dots ( b _ {s} ; q ) _ {n} }
 +
z  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846022.png" />. See [[#References|[a2]]].
+
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]]].
 
Hypergeometric functions of matrix argument have also been studied, cf. [[#References|[a3]]].

Revision as of 22:11, 5 June 2020


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.

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. [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 [a2].

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

References

[a1] P. Appell, M.J. Kampé de Fériet, "Fonctions hypergéométriques et hypersphériques: Polynômes d'Hermite" , Gauthier-Villars (1926)
[a2] G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press (1989)
[a3] K. Gross, D. Richards, "Special functions of matrix argument I" Trans. Amer. Math. Soc. , 301 (1987) pp. 781–811
How to Cite This Entry:
Hypergeometric series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypergeometric_series&oldid=47299
This article was adapted from an original article by E.A. Chistova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article