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arXiv · 2510.14124

An Orthogonal View of Gau\ss ian Polynomials

Christian Krattenthaler·Brandt Kronholm·Paul Marsh
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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 $\begin{bmatrix}N+4\\4\end{bmatrix}_q$.

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Christian Krattenthaler, Brandt Kronholm, Paul Marsh. 2025-10-15. An Orthogonal View of Gau\ss ian Polynomials. https://arxiv.org/abs/2510.14124

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