  We consider the heat equation $u_t=Lu$ where $L$ is a second-order difference
operator in a discrete variable $n$. The fundamental solution has an expansion
in terms of the Bessel functions of imaginary argument. The coefficients
$\alpha_k(n,m)$ in this expansion are analogs of Hadamard's coefficients for
the (continuous) Schrodinger operator.
  We derive an explicit formula for $\alpha_k$ in terms of the wave and the
adjoint wave functions of the Toda lattice hierarchy. As a first application of
this result, we prove that the values of these coefficients on the diagonals
$n=m$ and $n=m+1$ define a hierarchy of differential-difference equations which
is equivalent to the Toda lattice hierarchy. Using this fact and the
correspondence between commutative rings of difference operators and algebraic
curves we show that the fundamental solution can be summed up, giving a finite
formula involving only two Bessel functions with polynomial coefficients in the
time variable $t$, if and only if the operator $L$ belongs to the family of
bispectral operators constructed in [18].
