On a c-number quantum τ-function

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