Matrix elements of spinor and principal series representations of the Lorentz
group are studied in the basis of complex angular momentum (helicity basis). It
is shown that matrix elements are expressed via hyperspherical functions
(relativistic spherical functions). In essence, the hyperspherical functions
present itself a four-dimensional (with respect to a pseudo-euclidean metrics
of Minkowski spacetime) generalization of the usual three-dimensional spherical
functions. An explicit form of the hyperspherical functions is given. The
hypespherical functions of the spinor representations are represented by a
product of generalized spherical functions and Jacobi functions. It is shown
that zonal hyperspherical functions are expressed via the Appell functions. The
associated hyperspherical functions are defined as the functions on a
two-dimensional complex sphere. Integral representations, addition theorems,
symmetry and recurrence relations for hyperspherical functions are given. In
case of the principal and supplementary series representations of the Lorentz
group, the matrix elements are expressed via the functions represented by a
product of spherical and conical functions. The hyperspherical functions of the
principal series representations allow one to apply methods of harmonic
analysis on the Lorentz group. Different forms of expansions of square
integrable functions on the Lorentz group are studied. By way of example, an
expansion of the wave function, representing the Dirac field
$(1/2,0)\oplus(0,1/2)$, is given.