Structure of the 12j and 15j coefficients in the Bargmann approach

(A) The generating functions for the 12j and 15j coefficients are explicitly computed in the Bargmann scheme. It is a tribute to the powerful Bargmann lemmas on the Laplacian integrals4 that those seemingly complicated 6n-fold integrals can in fact be systematically carried out. Thus in prinCiple the generating functions for the 3nj coefficients are computable for arbitrary n in the Bargmann approach. Alternatively, the generating functions can also be found in the algebraic recursive scheme of Schwinger. 3 For n = 4 and 5, they have been verified; the answers are essentially the same apart from a difference in an over-all phase factor. 6