  The bosonic string theory evolved as an attempt to find physical/quantum
mechanical model capable of reproducing Euler's beta function (Veneziano
amplitude) and its multidimensional analogue. The multidimensional analogue of
beta function was studied mathematically for some time from different angles by
mathematicians such as Selberg, Weil and Deligne among many others. The results
of their studies apparently were not taken into account in physics literature
on string theory. In recent publication hep-th/0212189 [IJMPA 19 (2004) 1655]
an attempt was made to restore the missing links. The results of this
publication are incomplete, however, since no attempts were made at reproducing
known spectra of both open and closed bosonic string or at restoration of the
underlying model(s) reproducing such spectra. Nevertheless, discussed in this
publication the existing mathematical interpretation of the multidimensional
analogue of Euler's beta function as one of the periods associated with the
corresponding differential form "living" on the Fermat-type (hyper)surfaces,
happens to be crucial for restoration of the quantum/statistical mechanical
model reproducing such generalized beta function. Unlike the traditional
formulations, this new model is supersymmetric. Although details leading to the
restoration of this model are already presented in hep-th/0312294, the present
work is aimed at more focused exposition of some of earlier presented results
and is restricted mainly to the description of analytical properties of the
Veneziano and Veneziano-like amplitudes. As such, it constututes Part I of our
four parts work. Parts 2-4 will be devoted respectively to the group-theoretic,
symplectic and combinatorial treatments of this new string-like supersymmetric
model
