Guo–Zudilin q-analogue with full truncation

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Let (a;q)k=∏j=0k−1(1−aqj)(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j) be the qq-shifted factorial, let Φn(q)\Phi_n(q) be the nnth cyclotomic polynomial, and let n>1n>1 be odd and r⩾1r\geqslant 1. Guo–Zudilin's full-truncation q-conjecture. Modulo Φn(q)3\Phi_n(q)^3,

∑k=0nr+1−1(q;q2)k4(q2;q2)k4q2k≡(∑k=0n−1(q;q2)k4(q2;q2)k4q2k)(∑k=0nr−1(qn2;q2n2)k4(q2n2;q2n2)k4q2n2k).\sum_{k=0}^{n^{r+1}-1}\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k} \equiv \left(\sum_{k=0}^{n-1}\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k}\right) \left(\sum_{k=0}^{n^r-1}\frac{(q^{n^2};q^{2n^2})_k^4}{(q^{2n^2};q^{2n^2})_k^4}q^{2n^2k}\right).

The conjecture is a qq-analogue of a supercongruence whose verification is tied to the modular-form congruence of Van Hamme and Kilbourn. The source does not report a resolution.

References

Primary source

Victor J. W. Guo, “q-Analogues of Dwork-type supercongruences”, arXiv:1910.07551 (2019).

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