SPIN-MATRIX POLYNOMIALS AND THE VENEZIANO FORMULA

We consider a partial-wave expansion for $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\omega}$ in terms of the spin-matrix polynomials ${W}_{n}(z)=\frac{\ensuremath{\Gamma}(z+\frac{1}{2}n+\frac{1}{2})}{\ensuremath{\Gamma}(z\ensuremath{-}\frac{1}{2}n+\frac{1}{2})}$ rather than the conventional Legendre polynomials. This leads naturally to Veneziano's formula when his supplementary condition $\ensuremath{\alpha}(s)+\ensuremath{\alpha}(t)+\ensuremath{\alpha}(u)=2$ is invoked.