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We consider the heat equation $u_t=Lu$ where $L$ is a second-order differenceoperator in a discrete variable $n$. The fundamental solution has an expansionin terms of the Bessel functions of imaginary argument. The coefficients$\alpha_k(n,m)$ in this expansion are analogs of Hadamard's coefficients forthe (continuous) Schrodinger operator.  We derive an explicit formula for $\alpha_k$ in terms of the wave and theadjoint wave functions of the Toda lattice hierarchy. As a first application ofthis 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 whichis equivalent to the Toda lattice hierarchy. Using this fact and thecorrespondence between commutative rings of difference operators and algebraiccurves we show that the fundamental solution can be summed up, giving a finiteformula involving only two Bessel functions with polynomial coefficients in thetime variable $t$, if and only if the operator $L$ belongs to the family ofbispectral operators constructed in [18].Comment: corrected typo

Heat kernel expansions on the integers and the Toda lattice hierarchy