\begin{align*}
    a_i &= 1+\frac{1-p_B - p_W}{u+v+i} \\
    b_i &= \frac{p_B(v+i)+(1-p_W) u}{u+v+i} \\
    c_i &= -\frac{\left(1 -p_B- p_W\right){}^2}{(u+v+i)^2} \\
    d_i &= \frac{\left(p_B+p_W-1\right) \left(2 p_B (i+v)-i-2 u p_W+u-v\right)}{(u+v+i)^2} \\
    e_i &= \frac{\left(\left(1-p_B\right) (i+v)+u p_W\right) \left(p_B (i+v)+u \left(1-p_W\right)\right)}{(u+v+i)^2}.
\end{align*}
