  For well-defined Finsleroid-relativistic space $\cE_g^{SR}$ (with the
upperscript SR meaning Special-Relativistic) due only to accounting a
characteristic parameter $g$ which measures the deviation of the geometry from
its pseudoeuclidean precursor, the creation of the respective quantization
programs for relativistic physical fields seems to be an urgent task. The
parameter may take on the values over all the real range; at $g=0$ the space is
reduced to become an ordinary pseudoeuclidean one. In the present work, the
formulation of theory for relativistic physical fields in such a space is
initiated. A general method to solve respective scalar, electromagnetic, and
spinor field equations is proposed basing on the conformal flatness. At any
value of the parameter, the expansion of the relativistic fields with respect
to non-plane waves appeared is found, which proposes a base upon which the
fields can be quantized in context of the Finsleroid-relativistic approach.
Remarkably, the regulators can naturally be proposed to overcome divergences in
relativistic field integrals. The respective key and basic concepts involved
are presented. For short, the abbreviations FMF, FMT, and FHF will be used for
the Finsleroid-relativistic metric function, the associated metric tensor, and
the associated Hamiltonian function, respectively.
