Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations

Takehiro Hasegawa, Hayato Saigo, Seiken Saito, Shingo Sugiyama

TL;DR

The paper develops a unified framework for lattice sums of $I$-Bessel functions on sublattices, connecting these sums to theta-type inversion formulas and discrete heat kernels. It introduces a twisted Poisson summation approach to handle non-continuous test functions, then extends the identities to lattices with Dirichlet characters and derives character-based theta transformations, including eta-type results. It further links $I$-Bessel lattice sums to linear codes over rings via complete weight enumerators and MacWilliams-type identities, and culminates in explicit heat-kernel formulas for general lattices, enabling closed-form solutions of the heat equation on $\\mathbb{Z}^n$ with code-based initial data. The results unify discrete Bessel sums, coding theory, and modular forms, with potential applications to spectral zeta functions and lattice-based diffusion processes.

Abstract

We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by $I$-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of $I$-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that $I$-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on $\mathbb{Z}^n$ whose initial condition is given by a linear code.

Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations

TL;DR

The paper develops a unified framework for lattice sums of -Bessel functions on sublattices, connecting these sums to theta-type inversion formulas and discrete heat kernels. It introduces a twisted Poisson summation approach to handle non-continuous test functions, then extends the identities to lattices with Dirichlet characters and derives character-based theta transformations, including eta-type results. It further links -Bessel lattice sums to linear codes over rings via complete weight enumerators and MacWilliams-type identities, and culminates in explicit heat-kernel formulas for general lattices, enabling closed-form solutions of the heat equation on with code-based initial data. The results unify discrete Bessel sums, coding theory, and modular forms, with potential applications to spectral zeta functions and lattice-based diffusion processes.

Abstract

We extend a certain type of identities on sums of -Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by -Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of -Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that -Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on whose initial condition is given by a linear code.
Paper Structure (5 sections, 15 theorems, 98 equations)

This paper contains 5 sections, 15 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Formulas of lattice sums of $I$-Bessel functions
  3. Generalizations of sums of the $I$-Bessel functions on lattices with characters and theta transformation formulas
  4. Lattice sums of $I$-Bessel functions coming from linear codes
  5. Sums of $I$-Bessel functions and heat equations on lattices

Key Result

Theorem 1.1

Set $\chi((a_1,\dots, a_n)):=\prod_{j=1}^n \chi_j(a_j)$, where each $\chi_j$ is a primitive Dirichlet character modulo $q$. Let $\Gamma=\mathbf{Z}^nA$ with $A\in {\rm GL}_n(\mathbf{R})$ be a sublattice of $\mathbf{Z}^n$. Suppose that every entry of $A$ is divisible by $q$. Then, for any $(t_1,\dots, where each $\chi_j$ is regarded as a function on $\mathbf{Z}$ by ${\chi_j}(a)=0$ if $\gcd(a,q)\neq

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Theorem 2.6
  • Example 2.7
  • ...and 18 more