Possibility of an Infinite Sequence of Regge Recurrences

We show that a Regge trajectory, $\alpha \mathrm{}\left(s\right)\mathrm{}$, cannot have the property $\mathrm{Re}\alpha \mathrm{}\left(s\right)\mathrm{}\to +\infty$ as $s\to +\infty$ without leading to inconsistencies with two features of dispersion theory and Regge pole theory: that both $\alpha \mathrm{}\left(s\right)\mathrm{}$ and the reduced residue function, $\gamma \mathrm{}\left(s\right)\mathrm{}$, are analytic in the cut plane with one cut, and that they and the partial wave amplitude, $a(l, s)$, for $\mathrm{Re}l=-\frac{1}{2}$, are bounded for large $\mathrm{}\left|s\right|\mathrm{}$ by $\exp [{\mathrm{}\left|s\right|\mathrm{}}^{\frac{1}{2}}-\epsilon ]$.