Basic hypergeometric series

Remarks

For $r = s + 1$ the series ${}_r\phi_s$ converges for $\abs{z} < 1$; for $r \le s$ it converges for all $z \in \C$. If some $a_i = q^{-N}$ with an integer $N \ge 0$, then $(a_i; q)_n = 0$ for $n > N$, and the series terminates. The factor $\left( (-1)^n q^{\binom{n}{2}} \right)^{1+s-r}$ makes the limit $a_r \to \infty$ of ${}_r\phi_s(\dots; q, z/a_r)$ a series ${}_{r-1}\phi_s$.

Example: ${}_1\phi_0(a; -; q, z) = \sum_{n \ge 0} \frac{(a; q)_n}{(q; q)_n} z^n$, and here the ratio of the terms of index $n+1$ and $n$ is $\frac{1 - a q^n}{1 - q^{n+1}}\, z$, a rational function of $q^n$. By the q-binomial theorem it equals $\frac{(az; q)_\infty}{(z; q)_\infty}$ for $\abs{z} < 1$.

Non-example: $\sum_{n \ge 0} \frac{z^n}{n!}$ is a hypergeometric series but not a basic one, since the ratio $\frac{z}{n+1}$ is not a rational function of $q^n$.

As $q \to 1$, $\frac{(q^a; q)_n}{(1 - q)^n}$ tends to the Pochhammer symbol $(a)_n$, so ${}_{s+1}\phi_s(q^{a_1}, \dots; q^{b_1}, \dots; q, z)$ tends to ${}_{s+1}F_s(a_1, \dots; b_1, \dots; z)$.