thetagamma

Thetagamma is a term that appears in some mathematical literature to denote a function or family of functions obtained by combining a Jacobi theta function with the gamma function. It is not a standard term with a universally agreed definition, and its exact meaning varies between authors.

Common interpretations treat thetagamma as the pointwise product θ(z; q) Γ(z), where θ denotes a Jacobi theta function and Γ is the Euler gamma function. In other treatments thetagamma may refer to a theta kernel modified by gamma weights, or to related integral transforms in which the gamma factor acts as a weight or normalization. Because the notation is not standardized, different sources may use different constructions under the same name.

Analytic properties of such a function depend on the chosen constituents. The theta factor contributes modular-like or quasi-periodic behavior in its variables, while the gamma factor introduces poles at non-positive integers and engenders meromorphic behavior. The resulting object thus inherits a mixture of periodicity-type features and gamma’s singularity structure, with specifics contingent on the exact definition adopted.

Usage of thetagamma is found in specialized contexts within analytic number theory, spectral theory, and mathematical physics, where researchers study the interaction between modular-type components and gamma-type singularities. Given the lack of a universal definition, readers should consult the original source to understand the intended form and properties in any given treatment.

See also: Jacobi theta function; Gamma function; modular forms; theta kernel.