High-energy string scattering amplitudes and signless Stirling number identity

We give a complete proof of a set of identities (7) proposed recently from calculation of high-energy string scattering amplitudes. These identities allow one to extract ratios among high-energy string scattering amplitudes in the fixed angle regime from high-energy amplitudes in the Regge regime. The proof is based on a signless Stirling number identity in combinatorial theory. The results are valid for arbitrary real values $L$ rather than only for $L=0,1$ proved previously. The identities for non-integer real value $L$ were recently shown to be realized in high-energy compactified string scattering amplitudes [He S., Lee J.C., Yang Y., arXiv: 1012.3158]. The parameter $L$ is related to the mass level of an excited string state and can take non-integer values for Kaluza–Klein modes.