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Based on a summation algorithm for Stirling number identity developedrecently, we discover that the ratios calculated previously among high energystring scattering amplitudes in the Gross regime (GR) can be extracted from theKummer function of the second kind. This function naturally shows up in theleading order of high energy string scattering amplitudes in the Regge regime(RR). As a result, the identity suggested by string theory calculation can berigorously proved by a totally different but sophisticated mathematical method.We conjecture and give evidences that the existence of these ratios in the RRpersists to all orders in the Regge expansion of high energy string scatteringamplitudes.

Kummer function and High energy String Scatterings