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The tri-pentagonal number theorem and related identities. (English) Zbl 1192.33001

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.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11B65 Binomial coefficients; factorials; \(q\)-identities

Citations:

Zbl 0985.11048

Software:

qMultiSum
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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–
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