qseries

Qseries is a term used in mathematics to denote a broad family of series in which each term involves a power of a fixed base q. In most usual contexts, 0<|q|<1, and a q-series is written as sum_{n≥0} a_n q^n. When a_n=1 for all n, the geometric series sum_{n≥0} q^n = 1/(1−q) is the simplest example. More generally, q-series appear as generating functions in combinatorics and number theory, encoding partitions, compositions, and other combinatorial statistics.

A central toolkit around q-series includes the q-Pochhammer symbol (q;q)_n = ∏_{k=1}^n (1−q^k) and the basic hypergeometric series _rφ_s(a_1,...,a_r; b_1,...,b_s; q, z). Identities among q-series, such as Euler's product for partition functions and the Rogers–Ramanujan identities, reveal deep connections with modular forms and representation theory.

Historically, q-series emerged from Euler and Jacobi in the 18th and 19th centuries and were developed extensively in the 20th century by Ramanujan and others. Today they play a key role in combinatorics, number theory, algebraic geometry, and mathematical physics, appearing in topics ranging from partition theory and modular forms to knot invariants and quantum groups.

See also q-series, basic hypergeometric series, partition function, and Ramanujan's identities.