Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov–Witten Theory

We prove two multivariate $q$-binomial identities conjectured by Bousseau, Brini and van Garrel [Geom. Topol. 28 (2024), 393–496, arXiv:2011.08830] which give generating series for Gromov–Witten invariants of two specific log Calabi–Yau surfaces. The key identity in all the proofs is Jackson's $q$-analogue of the Pfaff–Saalschütz summation formula from the theory of basic hypergeometric series.