q-shifted factorial

The q-analog of the Pochhammer symbol defined by (a;q)_k = { product_(j = 0)^(k - 1)(1 - a q^j) | if k>0 1 | if k = 0 product_(j = 1)^( left bracketing bar k right bracketing bar ) (1 - a q^(-j))^(-1) | if k<0 product_(j = 0)^∞(1 - a q^j) | if k = ∞ auto right match (Koepf 1998, p. 25). q-Pochhammer symbols are frequently called q-series and, for brevity, (a;q)_n is often simply written (a)_n. Note that this contention has the slightly curious side-effect that the argument is not taken literally, so for example (-q)_n means (-q;q)_n, not (-q; - q)_n (cf. Andrews 1986b).

Borwein conjectures | Dedekind eta function | Fine's equation | Jackson's identity | Jacobi identities | mock theta function | Pochhammer symbol | q-analog | q-binomial coefficient | q-binomial theorem | q-cosine | q-factorial | Q-function | q-gamma function | q-hypergeometric function | q-multinomial coefficient | q-series | q-series identities | q-sine | Ramanujan Ψ sum | Ramanujan theta functions | Rogers-Ramanujan identities

QPochhammer

Leo August Pochhammer