Extensions of the truncated pentagonal number theorem
John M. Campbell
TL;DR
This paper addresses extending truncations of Euler's pentagonal number theorem by developing a systematic $q$-Zeilberger–based method to obtain finite-sum expansions for generalized truncated $q$-series of the form $1 + (-1)^{k-1} \sum_{n\ge k} \frac{ q^{\binom{k}{2}+(k+m)n} }{(q;q)_n} [ n - \ell_1 k - 1 ]_{q}$ with linear $p(k)$ and natural $\ell_1,\ell_2$, thereby unifying and generalizing the Andrews–Merca, Kolitsch, Yao, Schlosser–Zhou, and Xia–Yee–Zhao identities. It yields new truncation identities, including a new truncation for $(p(k),\ell_1,\ell_2)=(k+1,2,0)$, and extends to infinite families with $\rho(k)$ linear (e.g., $\rho(k)=2k+m, 3k+m, 4k+m$) with explicit $p(j)$ and $q$-hypergeometric factors; these results accommodate complex $m$ as well. The approach employs $q$-difference recurrences and telescoping to convert sums into finite expressions, and in the $k\to\infty$ limit reproduces unilateral forms of the Euler pentagonal theorem. Overall, the work provides a unifying framework for truncations of truncated theta-series and points to combinatorial interpretations and broader generalizations as future directions.
Abstract
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.