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Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov–Witten Theory
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Looijenga pairs, Gromov–Witten invariants, basic hypergeometric series
Mots-clés : log Calabi–Yau surfaces, $q$-binomial coefficients, Pfaff–Saalschütz summation formula. 
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     author = {Christian Krattenthaler},
     title = {Proof of {Two} {Multivariate} $q${-Binomial} {Sums} {Arising} in {Gromov{\textendash}Witten} {Theory}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a88/}
}
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Christian Krattenthaler. Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov–Witten Theory. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a88/

[1] Bousseau P., Brini A., van Garrel M., “Stable maps to Looijenga pairs”, Geom. Topol., 28 (2024), 393–496, arXiv: 2011.08830 | DOI

[2] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI