The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index
Abstract
For a nonnegative integer k and a rational number r∈ℚ+, we define the generalized Gaussian binomial coefficient qbinomr+kk = ((qr+1; q)k)/((q; q)k). When r=a/b with a,b coprime positive integers and b≥ 2, expanding qbinomr+kk via the finite q-binomial theorem produces fractional powers of q, so that qbinomr+kk is a \emph{Puiseux series} in q with nonnegative exponents; concretely it lies in ℚ[[q1/b]]. The notion we single out is the \emph{integer trace} of this expansion, the subseries consisting of those terms cr(d) qd whose exponent d is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as r varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{1/2-Conjecture}: among all r∈ℚ+, the value r=1/2 maximizes the integer trace, in the sense that the coefficients of qbinom1/2+kk dominate those of qbinomr+kk coefficientwise for every r. That so elementary a definition should single out 1/2 this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.
Related
- Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
- The Sufficient Set of Congruences of Binomial Coefficients for Primality
- Infinitely Many Binomial Coefficients of Deficiency One
- Binomial coefficients and the operadic butterfly
- Multiple harmonic star sums involving central binomial coefficients as weights
BibTeX
@misc{han202612conjecture,
title = {{The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index}},
author = {Guo-Niu Han and Huan Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.01919},
eprint = {2606.01919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01919},
}
RIS
TY - UNPB
TI - The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index
AU - Guo-Niu Han
AU - Huan Xiong
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2606.01919
UR - https://arxiv.org/abs/2606.01919
AB - For a nonnegative integer k and a rational number r∈ℚ^+, we define the generalized Gaussian binomial coefficient qbinomr+kk = ((q^r+1; q)_k)/((q; q)_k). When r=a/b with a,b coprime positive integers and b≥ 2, expanding qbinomr+kk via the finite q-binomial theorem produces fractional powers of q, so that qbinomr+kk is a \emph{Puiseux series} in q with nonnegative exponents; concretely it lies in ℚ[[q^1/b]]. The notion we single out is the \emph{integer trace} of this expansion, the subseries consisting of those terms c_r(d) q^d whose exponent d is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as r varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{1/2-Conjecture}: among all r∈ℚ^+, the value r=1/2 maximizes the integer trace, in the sense that the coefficients of qbinom1/2+kk dominate those of qbinomr+kk coefficientwise for every r. That so elementary a definition should single out 1/2 this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.
ER -