The discrete analogue of the Gaussian
This paper illustrates the utility of the heat kernel on Z as the discrete analogue of the Gaussian density function. The heat kernel on Z is the two-variable function KZ(t,x)=e−2tIx(2t) where Ix(2t) is a Bessel function, with variables x∈Z and t⩾0. Like its classic counterpart, KZ(t,x) appears in many mathematical and physical contexts and has a wealth of applications. Some of these applications will be reviewed here, and they concern Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how e−2tIx(2t) could be useful in differential privacy.