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-Saalsch"utz summation formula from the theory of basic hypergeometric series.