  We present a proof of the explicit formula for the asymptotically free
eigenfunctions of the $B_N$ $q$-Toda operator which was conjectured by the
first and third authors. This formula can be regarded as a branching formula
from the $B_N$ $q$-Toda eigenfunction restricted to the $A_{N-1}$ $q$-Toda
eigenfunctions. The proof is given by a contigulation relation of the $A_{N-1}$
Toda eigenfunctions and a recursion relation of the branching coefficients.
