Asymmetric Rogers-Ramanujan type identities. I: The Andrews-Uncu conjecture.(English)Zbl 1523.11188
In this article, the author establishes two single-symmetric Rogers-Ramanujan type identities,
(i). \(\sum_{n\ge0}\frac{(-1)^{n}q^{3\binom{n}{2}+4n}(q;q^{3})_{n}}{(q^{9};q^{9})_{n}}=\frac{(q^{4};q^{6})_{\infty}(q^{12};q^{18})_{\infty}}{(q^{5};q^{6})_{\infty}(q^{9};q^{18})_{\infty}}\), and
(ii). \(\sum_{n\ge0}\frac{(a;q)_{n}(a^{-1}q^{2};q^{2})_{n}}{(a^{2}q;q^{2})_{n}(q^{3};q^{3})_{n}}(-1)^{n}a^{n}q^{\binom{n}{2}+n} = \frac{(aq;q^{2})_{\infty}(a^{3}q^{3};q^{6})_{\infty}}{(a^{2}q;q^{2})_{\infty}(q^{3};q^{6})_{\infty}}\).
Further, using identity (i) as a key ingredient, and properties of contour integral and 3-dissections, the author proves the Andrews-Uncu Conjecture [G. E. Andrews andA. K. Uncu, Ramanujan J. 61, 715–729 (2023;doi:10.1007/s11139-022-00685-y)].
(i). \(\sum_{n\ge0}\frac{(-1)^{n}q^{3\binom{n}{2}+4n}(q;q^{3})_{n}}{(q^{9};q^{9})_{n}}=\frac{(q^{4};q^{6})_{\infty}(q^{12};q^{18})_{\infty}}{(q^{5};q^{6})_{\infty}(q^{9};q^{18})_{\infty}}\), and
(ii). \(\sum_{n\ge0}\frac{(a;q)_{n}(a^{-1}q^{2};q^{2})_{n}}{(a^{2}q;q^{2})_{n}(q^{3};q^{3})_{n}}(-1)^{n}a^{n}q^{\binom{n}{2}+n} = \frac{(aq;q^{2})_{\infty}(a^{3}q^{3};q^{6})_{\infty}}{(a^{2}q;q^{2})_{\infty}(q^{3};q^{6})_{\infty}}\).
Further, using identity (i) as a key ingredient, and properties of contour integral and 3-dissections, the author proves the Andrews-Uncu Conjecture [G. E. Andrews andA. K. Uncu, Ramanujan J. 61, 715–729 (2023;doi:10.1007/s11139-022-00685-y)].
Reviewer: M. P. Chaudhary (New Delhi)
MSC:
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 05A17 | Combinatorial aspects of partitions of integers |
| 33D60 | Basic hypergeometric integrals and functions defined by them |
Software:
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References:
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