  We discover an infinite number of recurrence relations among Regge string
scattering amplitudes \cite{bosonic,RRsusy} of different string states at
arbitrary mass levels in the open bosonic string theory. As a result, all Regge
string scattering amplitudes can be algebraically solved up to multiplicative
factors. Instead of decoupling zero-norm states in the fixed angle regime, the
calculation is based on recurrence relations and addition theorem of Kummer
functions of the second kind. These recurrence relations among Regge string
scattering amplitudes are dual to linear relations or symmetries among
high-energy fixed angle string scattering amplitudes discovered previously.
