Derivation of delta sin sub p for T=Z

Using the properties of $e_p$, it is clear that $\sin_p(t,s) = \dfrac{1-1}{2i} = 0$. Furthermore if $t>s$, then $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i}.$$ If $t<s$ then $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i}.$$