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Positivity of the basic hypergeometric series and the MacMahon identity
Abstract
The basic hypergeometric series n+1ϕn whose upper and lower parameters are non-negative integer powers of q satisfying a certain constraint can be, after a normalisation, computed as a positive polynomial. We derive three formulas for this polynomial. The first formula is a generalisation of the MacMahon identity in which an infinite binomial series is identified with a normalised generating function of des and maj statistics of multiset permutations. The second formula is a generating function of multiset permutations for the inversion statistic and a generalisation of the statistic mstc. The third formula is a finite sum containing 2n q-binomial coefficients.
Related
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BibTeX
@misc{garbali2026positivity,
title = {{Positivity of the basic hypergeometric series and the MacMahon identity}},
author = {Alexandr Garbali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07549},
eprint = {2610.07549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07549},
}
RIS
TY - UNPB
TI - Positivity of the basic hypergeometric series and the MacMahon identity
AU - Alexandr Garbali
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07549
UR - https://arxiv.org/abs/2610.07549
AB - The basic hypergeometric series _n+1ϕ_n whose upper and lower parameters are non-negative integer powers of q satisfying a certain constraint can be, after a normalisation, computed as a positive polynomial. We derive three formulas for this polynomial. The first formula is a generalisation of the MacMahon identity in which an infinite binomial series is identified with a normalised generating function of des and maj statistics of multiset permutations. The second formula is a generating function of multiset permutations for the inversion statistic and a generalisation of the statistic mstc. The third formula is a finite sum containing 2n q-binomial coefficients.
ER -