A double-strength q-congruence for truncated basic hypergeometric sums

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Let nn be a positive integer with n≡3(mod4)n\equiv 3\pmod{4}, let rr be any positive integer, and let Φn(q)\Phi_n(q) denote the nnth cyclotomic polynomial. Then the conjectured q-congruence.

∑k=0rn−1(q;q2)k2(q2;q4)k(q2;q2)k2(q4;q4)kq2k≡0(modΦn(q)2),\sum_{k=0}^{rn-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 0\pmod{\Phi_n(q)^2},

and

∑k=0rn+(n−1)/2(q;q2)k2(q2;q4)k(q2;q2)k2(q4;q4)kq2k≡0(modΦn(q)2).\sum_{k=0}^{rn+(n-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 0\pmod{\Phi_n(q)^2}.

This extends the preceding results for truncated basic hypergeometric sums in the case n≡3(mod4)n\equiv 3\pmod{4}; the asserted congruences modulo the square of the cyclotomic polynomial remain presented as conjectural here.

References

Primary source

Victor J. W. Guo and Wadim Zudilin, “On a q-deformation of modular forms”, arXiv:1812.11322 (2019).

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