# Analytic and arithmetic properties of the $(\Gamma,\chi)$-automorphic reproducing kernel function

## TL;DR

This paper explores the analytical and arithmetical properties of the $(3,chi)$-automorphic reproducing kernel function, focusing on zero distribution, lattice sums, and their connections to complex Hermite polynomials within the theta Bargmann-Fock space.

## Contribution

It introduces new results on the zero distribution, characterizes analytic sets, and generalizes arithmetic identities related to lattice functions using complex Hermite polynomials.

## Key findings

- Zeros of the kernel are discrete and finite within fundamental cells.
- Derived new lattice sums generalizing Perelomov's identities.
- Demonstrated the role of complex Hermite polynomials in lattice function analysis.

## Abstract

We consider the reproducing kernel function of the theta Bargmann-Fock Hilbert space associated to given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and discreteness of its zeros are examined and analytic sets inside a product of fundamental cells is characterized and shown to be finite and of cardinal less or equal to the dimension of the theta Bargmann-Fock Hilbert space. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite-Taylor coefficients. Some of them generalize some of the arithmetic identities established by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite-Taylor coefficients are nontrivial examples of the so-called lattice's functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.