A second q-analogue of the Swisher-type supercongruence

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Define the qq-shifted factorial by (a;q)0=1(a;q)_0=1 and (a;q)k=(1−a)(1−aq)⋯(1−aqk−1)(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1}) for k≥1k\geq 1, let Φn(q)\Phi_n(q) be the nn-th cyclotomic polynomial, and write [n]=1+q+⋯+qn−1[n]=1+q+\cdots+q^{n-1}. Let nn be a positive integer with n≡3  (mod⁡4)n\equiv 3\;\allowbreak(\operatorname{mod}4). The second q-analogue conjecture. Modulo Φn(q)2Φn2(q)2\Phi_n(q)^2\Phi_{n^2}(q)^2,

∑k=0(n2−1)/2(q;q2)k2(q2;q4)k(q2;q2)k2(q4;q4)kq2k≡[n2](q3;q4)(n2−1)/2(q5;q4)(n2−1)/2,\sum_{k=0}^{(n^2-1)/2}\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\equiv \frac{[n^2](q^3;q^4)_{(n^2-1)/2}}{(q^5;q^4)_{(n^2-1)/2}}, ∑k=0n2−1(q;q2)k2(q2;q4)k(q2;q2)k2(q4;q4)kq2k≡[n2](q3;q4)(n2−1)/2(q5;q4)(n2−1)/2.\sum_{k=0}^{n^2-1}\frac{(q;q^2)_k^2(q^2;q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{2k}\equiv \frac{[n^2](q^3;q^4)_{(n^2-1)/2}}{(q^5;q^4)_{(n^2-1)/2}}.

The conjecture is motivated by the preceding q-congruence and is proposed as a q-analogue of the two displayed Swisher-type congruences. The supplied text gives no resolution status.

References

Primary source

Victor J. W. Guo, “A family of q-congruences modulo the square of a cyclotomic polynomial”, arXiv:2001.08079 (2020).

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