Basic hypergeometric series

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, basic hypergeometric series, or q-hypergeometric series, are <a href="/facts/Q-analog/7jXz12AP">q-analogue</a> generalizations of <a href="/facts/Generalized_hypergeometric_series/fqXXDHdR">generalized hypergeometric series</a>, and are in turn generalized by <a href="/facts/Elliptic_hypergeometric_series/w3mEH4eT">elliptic hypergeometric series</a>. 
A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a> of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base. 
The basic hypergeometric series

2
 
 
 
 ϕ
 
 1
 
 
 (
 
 q
 
 α
 
 
 ,
 
 q
 
 β
 
 
 ;
 
 q
 
 γ
 
 
 ;
 q
 ,
 x
 )
 
 
 {\displaystyle {}_{2}\phi _{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)}
 
 was first considered by <a href="/facts/Eduard_Heine/FY8enC39">Eduard Heine</a> (1846). It becomes the hypergeometric series 
 
 
 
 F
 (
 α
 ,
 β
 ;
 γ
 ;
 x
 )
 
 
 {\displaystyle F(\alpha ,\beta ;\gamma ;x)}
 
 in the limit when base 
 
 
 
 q
 =
 1
 
 
 {\displaystyle q=1}
 
.

Basic hypergeometric series open-in-new

Basic hypergeometric series