DEGENERATE REPRESENTATION FUNCTIONS FOR SO,NU), SU(NU), AND SU(NU) BY SU(NU) AND THEIR ANALYTICAL REDUCTIONS

We have studied the problem of obtaining the rotation matrix elements dN(θ) = 〈N| e-iθJ2 |N〉, where | N〉 refer to a particular degenerate class of basis vectors of a symmetry group G which embraces the rotations SU(2)J as a subgroup. For G = SO(ν), SU(ν), and SU(ν) ⊗ SU(ν), we prove that these particular representation functions are proportional to the Gegenbauer polynomials CN1/2(ν-2)(cos θ), (C N1/2(ν-1)(cos θ), and CN 1/2ν(cos θ), respectively. The reduction of such functions into one another according to the formula CN′ λ′', = ΣNaNC Nλ has been solved in generality for complex values of N and corresponds to the reduction of Regge poles of G into Regge poles of one of its subgroups. The reduction formula for functions of the second type EN′λ′, = ΣNb NENλ has also been derived; here one simply meets an infinite series. Copyright © 1969 by the American Institute of Physics.