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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.
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