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

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