Orthonormal basis of hyperspherical harmonics for three identical particles

The basis of hyperspherical harmonics with quantum numbers of the "$SU(3)$ scheme" constructed earlier in the required form is analytically orthonormalized by means of an obtained recursion relation of the $L-1\to L$ type for the overlap integrals of the basis hyperspherical harmonics. The recursion relations and explicit expressions obtained earlier for calculating the basis hyperspherical harmonics are also represented in a simpler form.