截断五角数定理的扩展
Extensions of the truncated pentagonal number theorem
AI总结：
本文基于Zeilberger算法的$q$-版本提出一种方法，可获得一类$q$-级数的有限和展开，从而推广了Andrews-Merca及Kolitsch等人的截断五角数定理恒等式。
摘要：
Andrews和Merca为Euler五角数定理中$q$-级数的部分和引入并证明了一个$q$-级数展开。Kolitsch在2022年通过对正整数$m$给出$\sum_{n \geq k} \frac{ q^{ (k + m) n } }{ \left( q; q \right)_{n} } \left[ \begin{smallmatrix} n - 1 \\\\ k - 1 \end{smallmatrix} \right]_{q}$的有限和表达式，引入了Andrews-Merca恒等式的一种推广。Yao在2022年也证明了这个$q$-级数的一个等价求值，Schlosser和Zhou在2024年将此结果扩展到复数值$m$，其中$m = 1$的情形给出Andrews-Merca恒等式，$m = 2$的情形已由Xia、Yee和Zhao分别证明。我们引入并应用了一种基于Zeilberger算法$q$-版本的方法，可用于获得形如$\sum_{n \geq 1} \frac{ q^{ p(k) n } }{ \left( q; q \right)_{n + \ell_2} } \left[ \begin{smallmatrix} n - \ell_{1} \\\\ k - 1 \end{smallmatrix} \right]_{q}$的$q$-级数的有限和展开，其中$p(k)$为线性多项式，$\ell_{1} \in \mathbb{N}$且$\ell_{2} \in \mathbb{N}_{0}$，从而推广了Andrews-Merca恒等式以及Kolitsch、Yao和Schlosser-Zhou恒等式。例如，$(p(k), \ell_1, \ell_2) = (k+1, 2, 0)$的情形为Euler五角数定理提供了一个新的截断恒等式。
英文摘要：
Andrews and Merca introduced and proved a $q$-series expansion for the partial sums of the $q$-series in Euler's pentagonal number theorem. Kolitsch, in 2022, introduced a generalization of the Andrews-Merca identity via a finite sum expression for $ \sum_{n \geq k} \frac{ q^{ (k + m) n } }{ \left( q; q \right)_{n} } \left[ \begin{smallmatrix} n - 1 \\ k - 1 \end{smallmatrix} \right]_{q}$ for positive integers $m$, and Yao also proved an equivalent evaluation for this $q$-series in 2022, and Schlosser and Zhou extended this result for complex values $m$ in 2024, with the $m = 1$ case yielding the Andrews-Merca identity, and with the $m = 2$ case having been proved separately by Xia, Yee, and Zhao. We introduce and apply a method, based on the $q$-version of Zeilberger's algorithm, that may be used to obtain finite sum expansions for $q$-series of the form $ \sum_{n \geq 1} \frac{ q^{ p(k) n } }{ \left( q; q \right)_{n + \ell_2} } \left[ \begin{smallmatrix} n - \ell_{1} \\ k - 1 \end{smallmatrix} \right]_{q} $ for linear polynomials $p(k)$ and $\ell_{1} \in \mathbb{N}$ and $\ell_{2} \in \mathbb{N}_{0}$, thereby generalizing the Andrews-Merca identity and the Kolitsch, Yao, and Schlosser-Zhou identities. For example, the $(p(k), \ell_1, \ell_2) = (k+1, 2, 0)$ case provides a new truncation identity for Euler's pentagonal number theorem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
