 Veneziano amplitude pole residue

In units ${\alpha }^{\prime }=1$, generic fixed $t$ gives simple poles at $s=n-1$, $n\ge 0$, with residue $-{\prod }_{j=2}^{n+1}(t+j)/n!$. The empty product is one. A residue zero removes that pole, and fixed $t$ must be away from its own channel singularities.