An Orthogonal View of Gaußian Polynomials
Christian Krattenthaler, Brandt Kronholm, Paul Marsh
TL;DR
The Gaussian polynomials $[N+mm]_q$ have coefficient sequences with rich combinatorial structure, including unimodality and intricate relations among neighboring coefficients. The authors introduce perpendicular generating functions that reinterpret these coefficients via Gupta's partition technique and Ehrhart theory, using $s$-dissections and contour methods to obtain rational generating functions whose poles are roots of unity, hence yielding quasipolynomial coefficient behavior. They prove unimodality for $m=0$ through $6$, derive extensive difference identities for partitions with bounded largest part or number of parts, and establish Ramanujan-type congruences, complemented by explicit even/odd $m$ formulas and a Mathematica notebook that computes perpendicular functions up to $m=12$. The work provides new combinatorial tools linking unimodality, partition theory, and modular phenomena, with potential implications for representation theory and related identities in Lie theory. Overall, the paper expands the toolkit for studying Gaussian coefficients and their coefficient distributions, offering new proofs, identities, and computational methods that extend to all $m$ via the presented framework.
Abstract
We establish an alternative, ``perpendicular" collection of generating functions for the coefficients of Gaussian polynomials, $\begin{bmatrix}N+m\\m\end{bmatrix}_q$. We provide a general characterization of these perpendicular generating functions. For small values of $m$, unimodality of the coefficients of Gaussian polynomials is easily proved from these generating functions. Additionally, we uncover new and surprising identities for the differences of Gaussian polynomial coefficients, including a very unexpected infinite family of congruences for coefficients of $\begin{bmatrix}N+4\\4\end{bmatrix}_q$.