Degenerate Representation Functions for SO(v), SU(v), and SU(ν)⊗SU(ν) and Their Analytical Reductions
R. Delbourgo, K. Koller, Ruth M. Williams · Journal of Mathematical Physics · 1969
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(v), SU(v), and SU(ν)⊗SU(ν), we prove that these particular representation functions are proportional to the Gegenbauer polynomials CN12(ν−2)(cosθ),CN12(ν−1)(cosθ), and CN12ν(cosθ), respectively. The reduction of such functions into one another according to the formula CN′λ′=ΣNaNCNλ 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′=ΣNbNENλ has also been derived; here one simply meets an infinite series.