A q-analogue of Sun's supercongruence relating two truncated sums

Let nn be a positive odd integer, let qq be an indeterminate, let Φn(q)\Phi_n(q) be the nnth cyclotomic polynomial, and use (x;q)k=j=0k1(1xqj)(x;q)_k=\prod_{j=0}^{k-1}(1-xq^j) and [2kk]=(q;q)2k(q;q)k2\displaystyle {2k\brack k}=\frac{(q;q)_{2k}}{(q;q)_k^2}. q-analogue of Sun's supercongruence. One should have

k=0n1qk(q;q)k[2kk]k=0n1(q;q2)k2(q2;q2)k2q2k(modΦn(q)3).\sum_{k=0}^{n-1}\frac{q^k}{(-q;q)_k}{2k\brack k} \equiv \sum_{k=0}^{n-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{2k} \pmod{\Phi_n(q)^3}.

This is proposed as a qq-analogue of a known congruence for odd primes, motivated by related results of Guo, Zeng, Pan, Zhang, and Tauraso. The paper states that no such qq-analogue was known, even conjecturally, before this proposal.

Sources & referencesView supporting material

Primary source

Victor J. W. Guo, “q-Analogues of some supercongruences related to Euler numbers”, arXiv:2010.13526 (2020).

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