An Orthogonal View of Gaußian Polynomials

Abstract

We establish an alternative, ``perpendicular" collection of generating functions for the coefficients of Gaussian polynomials, ${\left[\begin{array}{c}N+m\\ m\end{array}\right]}_q$. 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 ${\left[\begin{array}{c}N+4\\ 4\end{array}\right]}_q$.

AmS-LaTeX, 26 pages; minor modifications