  We study a family of power series characterized by a system of recursion
relations (Q-system) with a certain convergence property. We show that the
coefficients of the series are expressed by the numbers which formally count
the off-diagonal solutions of the U_q(X^{(1)}_n) Bethe equation at q=0. The
series are conjectured to be the X_n-character of a certain family of
irreducible finite-dimensional U_q(X^{(1)}_n) -modules which we call the KR
(Kirillov-Reshetikhin) modules. Under the above conjecture, these coefficients
give a formula of the weight multiplicities of the tensor products of the KR
modules, which is also interpreted as the formal completeness of the XXZ-type
Bethe vectors.

Quantum Algebra