On the Weyl coefficients of SO(n)

It is shown that the representation functions of SO(n) can be obtained from those of SO(n−1) by calculating the Weyl coefficient of SO(n), which is equal to the representation function of SO(n) calculated at ϑ=π/2. Thus the problem of calculating the representation functions of SO(n) is reduced to its calculation at a particular value of ϑ (=π/2), instead of a whole range of values of ϑ. The Weyl coefficients of SO(3), SO(4), and SO(5) are explicitly obtained and discussed in detail. It is shown that the Weyl coefficients of SO(n) are all expressible as sums and products of 3-j symbols of SO(3) and normalization factorials. We also conclude that the representation functions of all orthogonal groups (as well as unitary groups) are ultimately reducible to the representation function of SO(2), i.e., eimϑ, multiplied by corresponding Weyl coefficients. Therefore, all representation functions of SO(n) and U(n) are ultimately expressible as Fourier series, whose coefficients can be explicitly calculated if one so wishes.