A q-supercongruence for a truncated Appell series when n is 3 modulo 4

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Let nn be a positive odd integer with n3(mod4)n\equiv 3\pmod{4}, let qq be an indeterminate, and let (a;q)k(a;q)_k denote the qq-shifted factorial. Write (a1,a2,,ar;q)k=s=1r(as;q)k(a_1,a_2,\ldots,a_r;q)_k=\prod_{s=1}^r(a_s;q)_k, and let Φn(q)\Phi_n(q) denote the nnth cyclotomic polynomial. The truncated Appell-series supercongruence.

i=0n12j=0n12(q;q2)i+j(q;q2)i(q;q2)j(q;q2)j(q2;q2)i2(q2;q2)j2(q2;q2)jq2i+2j0(modΦn(q)2).\sum_{i=0}^{\frac{n-1}{2}}\sum_{j=0}^{\frac{n-1}{2}}\frac{(q;q^{2})_{i+j}(q;q^{2})_{i}(q;q^{2})_{j}(-q;q^{2})_{j}}{(q^2;q^2)_i^2(q^2;q^2)_j^2(-q^2;q^2)_j}q^{2i+2j}\equiv0\pmod{\Phi_{n}(q)^2}.

This is proposed on the basis of numerical evidence as a qq-analogue of the first case of the paper's cited Appell-series result; its status is not resolved in the supplied text.

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Sources & referencesView supporting material

Primary source

He-Xia Ni, “Supercongruences Arising from Truncated Appell Series”, arXiv:2607.04583 (2026).

Additional references

10 papers in this index state this conjecture (2019–2026). The statement above is taken from the most recent of them; the others are arXiv:2201.04297, arXiv:2201.06942, arXiv:2103.06416, arXiv:2101.09753, arXiv:2012.13672, arXiv:2010.13526, arXiv:1912.00765, arXiv:1909.10294, arXiv:1901.07962.

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