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.

翻译：我们推广了一类先前由G. Chinta、J. Jorgenson、A Karlsson和M. Neuhauser给出的关于格点上$I$-Bessel函数求和恒等式。此外，我们证明在连续极限下，theta函数（如Dedekind eta函数）的变换公式可以借助带特征的$I$-Bessel格点求和恒等式导出。我们考虑了源于线性码的格点theta函数的类比，并证明由线性码定义的$I$-Bessel函数之和可用完全权重计数器表示。我们还证明$I$-Bessel格点求和作为一般格点上热方程的解出现。作为进一步应用，我们得到了$\mathbb{Z}^n$上热方程的一个显式解，其初始条件由线性码给出。