Equivalence of two functional equations for the shifted-diagonal tetrahedral-index generating function

Prove the equivalence, for all integers r and all nonzero complex z (in the domains of convergence), between the identities for the generating function varphi_r associated to shifted-diagonal tetrahedral-index series: (i) varphi_r(z q^{1/2}, q) · varphi_r(z q^{-1/2}, q) − varphi_{r+1}(z, q) · varphi_{r−1}(z, q) ≡ (z q^{-1/2})^{r} and (ii) varphi_r(z q, q) · varphi_r(z^{-1} q, q) − q^{r} z^{−r} · varphi_{r−2}(z, q) · varphi_{r+2}(z, q) ≡ 1.

Background

In Section 6 the authors introduce the generating functions varphi_r(z, q) that encode shifted-diagonal tetrahedral-index series and prove a key identity (Theorem 6.??) relating products of varphi_r at shifted arguments. They also propose an alternative functional equation that appears to capture the same structure but could not be shown equivalent to the first identity.

Establishing the equivalence would consolidate the analytic framework used to prove the initial cusp case and the Gang–Yonekura formula, revealing a deeper symmetry of the generating functions underpinning the 3D index and potentially simplifying subsequent arguments.