On summation of $3j$-Wigner symbols

Within the framework of the theory of irreducible tensor operators, using well-known general analytical results for double sums ($\sum_{jm}$) of products of two $3j$-Wigner symbols, analytical expressions for single sums ($\sum_m$) for the values $j_1 = j_2 = 1$ and $j = 2$ parameters of the upper row $3j$-Wigner symbol are specified. The expressions obtained supplement the well-known analytical results of the theory of angular momentum and are in demand in solving, in particular, such problems of atomic physics as the construction of a nonrelativistic quantum theory of single and double bremsstrahlung when a photon is scattered by an atom (atomic ion) (Hopersky et al [6,7,8]) and two-photon resonance single ionization of the deep shell of an atomic ion (Hopersky et al [9]).