Berkovich, Alexander The tri-pentagonal number theorem and related identities. (English) Zbl 1192.33001 Int. J. Number Theory 5, No. 8, 1385-1399 (2009). The author considers a proof of the pentagonal number theorem of G. E. Andrews [J. Comb. Theory, Ser. A 91, No. 1–2, 464–475 (2000; Zbl 0985.11048)]. He focuses on an identity that comes out of this proof: \[ \sum_{i,j,k} (-1)^{i+j+k} q^{\binom{i+j+k}{2}} \left[{2L}\atop{L-i}\right]_q\left[{2M}\atop{M-j}\right]_q\left[{2N}\atop{N-k}\right]_q= \frac{(q)_{2L}(q)_{2M}(q)_{2N}}{(q)_{L+M-N}(q)_{L+N-M}(q)_{M+N-L}}. \]First he reproves this from a simple polynomial identity. Then he uses this result to prove a variety of new formulas such as \[ \sum_{n_1,\dots,n_{k+1}\geq 0} \frac{q^{N_1^2+\dots+N_{k+1}^2}}{(q)_{n_1} \dots(q)_{n_k}(q^2;q^2)_{n_{k+1}} (q;q^2)_{n_k+n_{k+1}} } = \frac{(q^{4+6k};q^{4+6k})_\infty}{(q)_\infty}\frac{[q^{2+2k};q^{4+6k}]_\infty}{[q^{1+k};q^{4+6k}]_\infty} , \]for \(k\geq 0\), as well as new identities involving the tri-pentagonal theorem. Reviewer: Matilde Lalin (Montreal) Cited in 1 Document MSC: 33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\) 11B65 Binomial coefficients; factorials; \(q\)-identities Keywords:multidimensional \(q\)-binomial identities; quituple product identity; Bailey lemmas; rational function identities; recursion relations Citations:Zbl 0985.11048 Software:qMultiSum PDFBibTeX XMLCite \textit{A. Berkovich}, Int. J. Number Theory 5, No. 8, 1385--1399 (2009; Zbl 1192.33001) Full Text: DOI arXiv References: [1] DOI: 10.1006/jcta.2000.3111 · Zbl 0985.11048 · doi:10.1006/jcta.2000.3111 [2] DOI: 10.2140/pjm.1984.114.267 · Zbl 0547.10012 · doi:10.2140/pjm.1984.114.267 [3] DOI: 10.1007/978-0-387-78510-3_1 · Zbl 1183.11063 · doi:10.1007/978-0-387-78510-3_1 [4] DOI: 10.1090/S0002-9947-04-03680-3 · Zbl 1061.33015 · doi:10.1090/S0002-9947-04-03680-3 [5] DOI: 10.1016/0022-314X(83)90043-4 · Zbl 0516.10008 · doi:10.1016/0022-314X(83)90043-4 [6] DOI: 10.1017/CBO9780511526251 · doi:10.1017/CBO9780511526251 [7] DOI: 10.1016/0022-247X(85)90368-3 · Zbl 0582.10008 · doi:10.1016/0022-247X(85)90368-3 [8] DOI: 10.1016/S0747-7171(02)00138-4 · Zbl 1020.33007 · doi:10.1016/S0747-7171(02)00138-4 [9] Slater L. J., Proc. London Math. Soc. (2) 53 pp 460– [10] Slater L. J., Proc. London Math. Soc. (2) 54 pp 147– This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.