An Orthogonal View of Gaußian Polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912651425087488 |
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| author | Krattenthaler, Christian Kronholm, Brandt Marsh, Paul |
| author_facet | Krattenthaler, Christian Kronholm, Brandt Marsh, Paul |
| contents | 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 $\begin{bmatrix}N+4\\4\end{bmatrix}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Orthogonal View of Gaußian Polynomials Krattenthaler, Christian Kronholm, Brandt Marsh, Paul Number Theory Combinatorics Primary 11P81, Secondary 05A17, 05A15, 05A19 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 $\begin{bmatrix}N+4\\4\end{bmatrix}_q$. |
| title | An Orthogonal View of Gaußian Polynomials |
| topic | Number Theory Combinatorics Primary 11P81, Secondary 05A17, 05A15, 05A19 |
| url | https://arxiv.org/abs/2510.14124 |