A generalized q-supercongruence with shifted summation

Let pp be an odd prime, let m,rm,r be positive integers with pmp\nmid m, and let sp1s\leqslant p-1 be nonnegative. Write ap\langle a\rangle_p for the least nonnegative residue of aa modulo pp. The generalized q-supercongruence. If r/mps+1(mod2)\langle-r/m\rangle_p\equiv s+1\pmod2, then

k=sp1(qm;qm)2k(qr;qm)k(qmr;qm)kqmk(qm;qm)ks(qm;qm)k+s(q2m;q2m)k20(mod[p]2).\sum_{k=s}^{p-1}\frac{(q^m;q^m)_{2k}(q^r;q^m)_k(q^{m-r};q^m)_kq^{mk}}{(q^m;q^m)_{k-s}(q^m;q^m)_{k+s}(q^{2m};q^{2m})_k^2}\equiv0\pmod{[p]^2}.

This is presented as a further generalization suggested by the authors; the supplied text gives no resolution.

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Primary source

Victor J. W. Guo and Jiang Zeng, “Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas”, arXiv:1408.0512 (2014).

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