Lemma A2. Let Z = diag ( z$_{1}$, ..., z$_{r}$ ) , where z$_{1}$, ..., z$_{r}$ are complex variables, and let b$_{1}$, ..., b$_{p}$ be complex constants. Then The series on the right-hand side of this equality converges uniformly over Ω$_{ρ}$ = {Z : max$_{j}$$_{≤}$$_{r}$ | z$_{j}$ | ≤ ρ − $^{1}$} , for any ρ > max$_{s}$$_{≤}$$_{p}$ | b$_{s}$ | . Proof: Macdonald (1995, Chapter VI, § 10) shows that r ∏ j =1 p ∏ s =1 (1 − b$_{s}$z$_{j}$ ) − 1 /α = ∑ ∞ k =0 ∑ κ ⊢ k P ( α ) κ ( B ) Q ( α ) κ ( Z ) , where Q ( α ) κ = c ( κ,α ) c ′ ( κ,α $_{)}$P ( α ) κ . This result together with (35) imply (36). The uniform convergence in (36) follows from the fact that func- tion ∏ r j =1 ∏ p s $_{=1}$(1 − b$_{s}$z$_{j}$ ) − 1 /α is analytic in an open region that includes Ω$_{ρ}$. □ We are now ready to prove Lemma 1. Consider the right-hand side of (1), which we will denote as RHS. We will assume that max$_{s}$$_{≤}$$_{p}$ | b$_{s}$ | < 1 and that the contour K is the unit circle in the complex plane. That these assumptions are without loss of generality follows from the fact that the value of RHS does not change under the transformation Z → ϕ Z , B → ϕB, and A → ϕ − $^{1}$A, where ϕ is any positive num- ber, and under a deformation of K into the unit circle (because such a deformation leaves the contour in the region of the analyticity of the integrand). With these as- sumptions, and noting that the component ∏ r $_{j>i}$( z$_{j}$ − z$_{i}$ ) 2 /α of ω ( α $^{)}$( A, B, Z ) equals ( − 1) r ( r − 1) / (2 α ) ∏ r j =1 z ( r − 1) /α j ∏ r j = i ( 1 − z$_{i}$z − 1 j ) 1 /α , we can rewrite RHS for α = 2 /β, where β is any positive integer, in the form of the torus scalar product where RHS = γ ( α ) 〈 $_{0}$F ( α ) 0 ( A , Z ) , r ∏ j =1 ( z$_{j}$/a$_{j}$ ) ( p − r +1) /α − 1 r ∏ j =1 p ∏ s =1 (1 − b$_{s}$z$_{j}$ ) − 1 /α 〉$_{α}$ , γ ( α ) = r ∏ j =1 [ Γ (( p +1 − j ) /α ) Γ (1 /α ) Γ (( r +1 − j ) /α ) ] . (37) r ∏ j =1 p ∏ s =1 (1 − b$_{s}$z$_{j}$ ) − 1 /α = ∞ ∑ k =0 ∑ κ ⊢ k w ( κ, α ) ( α | κ | | κ | !) $_{2}$C ( α ) κ ( B ) C ( α ) κ ( Z ) . (36) 24