The $(C,A)$ inversion formula

Let $p,q,t\in\mathbb C$ satisfy $M<|t|^{n+1}<1$. For fixed $w\in\mathbb T^n$, let $C_w$ be a contour inside the annulus

$$\mathbb A=\{z\in\mathbb C\mid |t|-\epsilon<|z|<|t|^{-1}+\epsilon\}$$

for infinitesimally small positive $\epsilon$, with the points $t^{-1}w_j$ for $j\in\{1,\ldots,n+1\}$ in its interior, and put $C_w^n=C_w\times\cdots\times C_w$. Let $f$ be a $C_n$-symmetric function holomorphic on $\mathbb A^n$, and let $x\in\mathbb C^n$ satisfy $|t|<|x_j|<|t|^{-1}$. Here $\kappa^{\mathscr A}$ and $\kappa^{\mathscr C}$ are the normalization factors and $\Delta^{(\mathscr C,\mathscr A)}$ is the kernel displayed in the source. $(C,A)$ inversion formula. One has

$$\kappa^{\mathscr A}\kappa^{\mathscr C}\int_{\mathbb T^n}\left(\int_{C_w^n}\Delta^{(\mathscr C,\mathscr A)}(z,w,x;t)f(z)\,\frac{\mathrm dz}{z}\right)\frac{\mathrm dw}{w}=f(x),$$

where

$$\Delta^{(\mathscr C,\mathscr A)}(z,w,x;t)=\frac{\prod_{i=1}^n\prod_{j=1}^{n+1}\Gamma(tx_i^{\pm}w_j^{-1},t^{-1}z_i^{\pm}w_j)}{\prod_{i=1}^n\Gamma(z_i^{\pm2})\prod_{1\leq i<j\leq n}\Gamma(z_i^{\pm}z_j^{\pm})} \times\frac{1}{\prod_{1\leq i<j\leq n+1}\Gamma(w_iw_j^{-1},w_i^{-1}w_j,t^{-2}w_iw_j,t^2w_i^{-1}w_j^{-1})}.$$

The formula is proposed as a companion to the preceding inversion theorem for elliptic beta integrals; its status is not resolved in the supplied text.