  The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and
Forrester, can be viewed as a space of sequences (paths) {a_0,a_1,...,a_L},
with a_j-integers restricted by 1<=a_j<=\nu, |a_j-a_{j+1}|=1, a_0=s, a_L=r. In
this paper we introduce different basis which, as shown here, has the same
dimension as that of an RSOS-model. Following McCoy et al, we call this basis
-- fermionic (FB). Our first theorem Dim(FB)=Dim(RSOS-basis) can be succinctly
expressed in terms of some identities for binomial coefficients. Remarkably,
these binomial identities can be q-deformed. Here, we give a simple proof of
these q-binomial identities in the spirit of Schur's proof of the
Rogers-Ramanujan identities. Notably, the proof involves only the elementary
recurrences for the q-binomial coefficients and a few creative observations.
Finally, taking the limit L --> \infty in these q-identities, we derive an
expression for the character formulas of the unitary minimal series
M(\nu,\nu+1) "Bosonic Sum = Fermionic Sum". Here, Bosonic Sum denotes
Rocha-Caridi representation (\chi_{r,s=1}^{\nu,\nu+1}(q)) and Fermionic Sum
stands for the companion representation recently conjectured by the Stony Brook
group.
