Generalized Theta Functions. I

TL;DR

This paper introduces generalized theta functions with multiple parameters, explores their properties, differential equations, and representations, and discusses potential applications like embedding tori into projective space.

Contribution

It proposes a new class of generalized theta functions with multiple parameters and analyzes their properties, differential equations, and group representations.

Findings

Generalized theta functions include any even number of parameters.

Differential equations for these functions are derived.

Applications include projective embedding of tori.

Abstract

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential equations for generalized theta functions and finite non-unitary representations of extended Heisenberg group are presented so as other properties and possible applications are pointed out such as a projective embedding of tori by means of generalized theta functions.