Basic hypergeometric series

Summary

A basic hypergeometric series, or $q$-hypergeometric series, is a power series $\sum_n c_n z^n$ in which the ratio $c_{n+1}/c_n$ is a rational function of $q^n$. It is written ${}_r\phi_s(a_1, \dots, a_r; b_1, \dots, b_s; q, z)$ and is built from q-Pochhammer symbols. It is the q-analog of the generalized hypergeometric series, to which it tends as $q \to 1$ after a change of parameters. Basic hypergeometric series contain the partition generating functions, the theta products and the $q$-orthogonal polynomials of the Askey scheme.