Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums

The six gluon disk amplitude is calculated in superstring theory. This amplitude probes the gauge interactions with six external legs on Dp-branes, in particular including e.g. F^6-terms. The full string S-matrix can be expressed by six generalized multiple hypergeometric functions (triple hypergeometric functions), which in the effective action play an important role in arranging the higher order alpha' gauge interaction terms with six external legs (like F^6, D^4 F^4, D^2 F^5, D^6 F^4, D^2 F^6, ...). A systematic and efficient method is found to calculate tree-level string amplitudes by equating seemingly different expressions for one and the same string S-matrix: Comparable to Riemann identities appearing in string-loop calculations, we find an intriguing way of using world-sheet supersymmetry to generate a system of non-trivial equations for string tree-level amplitudes. These equations result in algebraic identities between different multiple hypergeometric functions. Their (six-dimensional) solution gives the ingredients of the string S-matrix. We derive material relevant for any open string six-point scattering process: relations between triple hypergeometric functions, their integral representations and their alpha'-(momentum)-expansions given by (generalized) Euler-Zagier sums or (related) Witten zeta-functions.