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 Zn whose
initial condition is given by a linear code