EXtheta

EXtheta is a mathematical construct described as a one-parameter extension of the classical theta function. It is used in contexts where a linear phase shift is imposed on lattice sums, providing a compact way to encode twisted boundary conditions and phase-modulated series.

Definition and basic properties: For a complex variable z in the upper half-plane and a real parameter theta, the EXtheta function is defined by EXtheta(z; theta) = sum over all integers n of exp(pi i n^2 z) times exp(i n theta). When theta equals zero, EXtheta reduces to a conventional theta function. The series is analytic in z for Im(z) > 0 and inherits many qualitative features of the ordinary theta function, including rapid decay under suitable transformations and a form of quasi-periodicity with respect to z. The presence of the linear phase theta introduces a theta-dependent multiplier, yielding modular-like transformation properties that depend on both z and theta.

Relation to existing theory: EXtheta connects to Jacobi theta functions and lattice sum techniques from number theory and mathematical physics. It often appears in analyses that use the Poisson summation formula with an added phase, enabling the study of twisted partition functions, spectral sums, and problems involving boundary conditions in systems with periodic structure. The concept emphasizes how small phase twists can alter symmetry properties while preserving core analytic structure.

History and terminology: The term EXtheta stands for an extended or exponential-modulated theta concept and has been used in theoretical discussions to distinguish phase-modulated theta sums from the standard theta function. It is treated as a generalization rather than a replacement, with various variants explored in specialized literature.

See also: Theta function, Jacobi theta, modular forms, Poisson summation, lattice sums.