The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index

Abstract

For a nonnegative integer $k$ and a rational number $r\in {\mathbb{Q}}^{+}$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\ge 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a \emph{Puiseux series} in $q$ with nonnegative exponents; concretely it lies in $\mathbb{Q}[[q^{1/b}]]$. 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{$\frac{1}{2}$-Conjecture}: among all $r\in {\mathbb{Q}}^{+}$, the value $r=\frac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\qbinom{1/2+k}{k}$ dominate those of $\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\frac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.