Lemma 38 ([Fixed-deficit coupling limit]). Suppose j₁,j_2→∞ and j₂/(j₁+j₂)→u² , where 0<u<1 . Put v=√1-u² . For fixed integers 0≤z≤T , the normalized coupled vector of spin j₁+j₂-z at total lowering deficit T can be chosen with …

At T=z , the raising equation gives the adjacent ratio -√((z-p)(2j_2-z+p+1))/((p+1)(2j_1-p)). With sign (-1)^z at p=0 , normalization yields the coefficients of (uX-vY)^z/√z! in the limit. The change of variables (X,Y)↦(uX-vY,vX+uY) is orthogonal, so these polynomials have norm one in the normalized-monomial inner product. Normalized lowering from T to T+1 divides the sum of the two lowering operators by √ (T-z+1) 2(j_1+j_2-z)-(T-z) . In the limit this acts by multiplication by (vX+uY)/√T-z+1 . Induction proves the statement for fixed descendants.