The Heine Transformation

E. Heine [178], [179, pp. 97–125] was the first to generalize Gauss’s hypergeometric series to q-hypergeometric series by defining, for $|q|<1$,

$$_2\phi_1\left(\begin{matrix} a, b\\c\end{matrix}; q,t\right) := \sum_{n=0}^{\i}\frac{(a;q)_n(b;q)_n}{(q;q)_n(c;q)_n}t^n,$$

((1.1.1))

, where $|t|<1$ and where, for each nonnegative integer n,

$$(a)_n = (a;q)_n := (1-a)(1-aq)\cdots(1-aq^{n-1}),$$

((1.1.2))

.