There are a great many beautiful identities involving q-series, some of which follow directly by taking the q-analog of standard combinatorial identities, e.g., the q-binomial theorem sum_(n = 0)^∞ ((a;q)_n z^n)/(q;q)_n = (a z;q)_∞/(z;q)_∞ ( left bracketing bar z right bracketing bar <1, left bracketing bar q right bracketing bar <1; Andrews 1986, p. 10), a special case of an identity due to Euler (a q;q)_∞ = sum_(k = 0)^∞ ((-1)^k q^(k(k + 1)/2) a^k)/(q;q)_k (Gasper and Rahman 1990, p.

Jacobi identities | q-hypergeometric function | q-series | q-Vandermonde sum | Ramanujan theta functions | Rogers-Ramanujan identities