  We present a ``natural finitization'' of the fermionic q-series (certain
generalizations of the Rogers-Ramanujan sums) which were recently conjectured
to be equal to Virasoro characters of the unitary minimal conformal field
theory (CFT) M(p,p+1). Within the quasi-particle interpretation of the
fermionic q-series this finitization amounts to introducing an ultraviolet
cutoff, which -- contrary to a lattice spacing -- does not modify the linear
dispersion relation. The resulting polynomials are conjectured (proven, for
p=3,4) to be equal to corner transfer matrix (CTM) sums which arise in the
computation of order parameters in regime III of the r=p+1 RSOS model of
Andrews, Baxter, and Forrester. Following Schur's proof of the Rogers-Ramanujan
identities, these authors have shown that the infinite-lattice limit of the CTM
sums gives what later became known as the Rocha-Caridi formula for the Virasoro
characters. Thus we provide a proof of the fermionic q-series representation
for the Virasoro characters for p=4 (the case p=3 is ``trivial''), in addition
to extending the remarkable connection between CFT and off-critical RSOS
models. We also discuss finitizations of the CFT modular-invariant partition
functions.
