On a $c$-number quantum $\tau $-function

We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne 1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.