Following the works of Schwinger (1965) and Bargmann (1962) on the three-dimensional rotation-group, the author constructs new generating functions for the Talmi coefficients (1952) and the Moshinsky-Smirnov coefficients (1952). Such a construction is achieved by manipulating a new generating function for the harmonic oscillator spherical basis. The latter generating function is constructed in turn from the well-known generating functions for the Laguerre polynomials and the spherical harmonics. The material is used to get new expressions for the Talmi and Moshinsky-Smirnov coefficients, which are particularly appropriate to hand and machine calculations. The author also mentions how particular relations both for the Talmi and the Moshinsky-Smironv coefficients can be obtained from the general expressions. Finally, new expressions, which are useful from a practical point of view, between Moshinsky-Smirnov coefficients are established.