Generalizations of the truncated pentagonal number theorem results (Q2675262)

\textit{G. E. Andrews} and the reviewer considered Euler's pentagonal number theorem \[ (q;q)_\infty = \sum_{n=-\infty}^\infty (-1)^n\, q^{n(3n-1)/2} \] and proved in [J. Comb. Theory, Ser. A 119, No. 8, 1639--1643 (2012; Zbl 1246.05014)] the following truncated form: For any $k\ge 1$, \[ \frac{(-1)^{k-1}}{(q;q)_\infty} \sum_{n=-(k-1)}^{k} (-1)^{n}\, q^{n(3n-1)/2}= (-1)^{k-1}+ \sum_{n=k}^\infty \frac{q^{\binom{k}{2}+(k+1)n}}{(q;q)_n} \begin{bmatrix} n-1\\ k-1 \end{bmatrix}, \] where $(a;q)_n=(a;q)_{\infty}/(aq^n;q)_{\infty}$, and \[ \begin{bmatrix} n\\ k \end{bmatrix} = \begin{cases} \dfrac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}, & \text{ if }0\leqslant k\leqslant n,\\ 0, &\text{ otherwise.} \end{cases} \] This result opened up a new study on truncated series. Since then, many truncated versions of theta series have been discovered. In this paper, results associated with the truncated pentagonal number theorem are generalized.