  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.
