  We present a slightly modified prescription of the radial pullback formalism
proposed previously by R. Manvelyan, R. Mkrtchyan and W. R\"uhl in 2012, where
authors investigated possibility to connect the main term of higher spin
interaction in flat $d+2$ dimensional space to the main term of interaction in
$AdS_{d+1}$ space ignoring all trace and divergent terms but expressed directly
through the $AdS$ covariant derivatives and including some curvature
corrections. In this paper we succeeded to solve all necessary \emph{recurrence
relations} to finalize full radial pullback of the main term of cubic
self-interaction for higher spin gauge fields in Fronsdal's formulation from
flat to one dimension less $AdS_{d+1}$ space. Nontrivial solutions of
recurrence relations lead to the possibility to obtain the full set of
$AdS_{d+1}$ dimensional interacting terms with all curvature corrections
including trace and divergence terms from any interaction term in $d+2$
dimensional flat space.
