  In this paper we study an approximation of tensor product of irreducible
integrable $\hat{\mathfrak{sl}_2}$ representations by infinite fusion products.
This gives an approximation of the corresponding coset theories. As an
application we represent characters of spaces of these theories as limits of
certain restricted Kostka polynomials. This leads to the bosonic (which is
known) and fermionic (which is new) formulas for the $\hat{\mathfrak{sl}_2}$
branching functions.
