$$\begin{array}{ll} \sin_p(t,s) &= \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} \\ &= \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} \end{array}$$
