Thetaq

Thetaq is a term used in mathematics to denote a family of theta-like objects parameterized by a base q. It refers to generalized, q-deformed theta constructs that interpolate between classical theta functions and broader q-series frameworks. In many treatments, thetaq is presented as a generating function or a functional object that reduces to familiar theta functions in certain limits or specializations of q. The precise definition and form of thetaq can vary between sources, reflecting different conventions for series, products, or integral representations.

Properties commonly associated with thetaq include features reminiscent of classical theta functions, such as symmetry under certain discrete transformations and connections to elliptic-type structures. Thetaq objects often serve as bridges between additive and multiplicative perspectives, linking combinatorial identities in q-series with analytic properties studied in number theory and representation theory. They also appear in mathematical physics, where deformations parameterized by q model lattice systems and other discretized or deformed theories.

Variants of thetaq may be expressed through infinite series, infinite products, or integral representations, each offering different analytic advantages. The deformation parameter q typically satisfies a constraint like |q| < 1 in analytic contexts, though alternative normalizations are used in other settings.

Origin and usage: The label thetaq is not tied to a single canonical definition and is used across sources to describe a class of q-deformed theta-like constructions. It serves as a convenient shorthand for exploring deformations of classical theta theory, modular-type behavior, and their applications.

See also: theta function, q-series, modular forms, Jacobi theta functions, elliptic functions.