\[(f_{0}+2\lambda_{j})sa_{s,j}=(s-\mu_{j})(s-1-\mu_{j})a_{s-1,j}+\sum_{r=1}^{s}%
\left(\lambda_{j}f_{r+1}+g_{r+1}-(s-r-\mu_{j})f_{r}\right)a_{s-r,j},\]