  The derivation scheme for hyperspherical harmonics (HSH) with arbitrary
arguments is proposed. It is demonstrated that HSH can be presented as the
product of HSH corresponding to spaces with lower dimensionality multiplied by
the orthogonal (Jacobi or Gegenbauer) polynomial. The relation of HSH to
quantum few-body problems is discussed. The explicit expressions for
orthonormal HSH in spaces with dimensions from 2 to 6 are given. The important
particular cases of four- and six-dimensional spaces are analyzed in detail and
explicit expressions for HSH are given for several choices of hyperangles. In
the six-dimensional space, HSH representing the kinetic energy operator
corresponding to i) the three-body problem in physical space and ii) four-body
planar problem are derived.
