Basic hypergeometric series

In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series x_n is called hypergeometric if the ratio of successive terms x_n+1/x_n is a rational function 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 ${\displaystyle {}_{2}\phi _{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)}$ was first considered by Eduard Heine (1846). It becomes the hypergeometric series ${\displaystyle F(\alpha ,\beta ;\gamma ;x)}$ in the limit when base ${\displaystyle q=1}$.