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

Christian Krattenthaler

Abstract

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.

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

Abstract

We prove two multivariate -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 -analogue of the Pfaff-Saalschütz summation formula from the theory of basic hypergeometric series.

Paper Structure

This paper contains 1 section, 3 theorems, 9 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

For integers $d_0$ and $d_1$ with $d_0>d_1\ge1$, we have

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Proposition 3