## Common $$q$$-analogues of some different supercongruences.(English)Zbl 1414.33016

Summary: We prove some $$q$$-congruences modulo the cube of a product of two cyclotomic polynomials by using Watson’s $$_8\phi _7$$ transformation formula and the creative microscoping method, recently devised by the author in collaboration with Wadim Zudilin. When $$n$$ is an odd prime power, we deduce different supercongruences from each of these $$q$$-congruences by taking the limit as $$q\rightarrow 1$$ or $$q\rightarrow -1$$. As a conclusion, we confirm the $$m=3$$ case of Conjecture 1.1 in [the author, Integral Transforms Spec. Funct. 28, No. 12, 888–899 (2017; Zbl 1379.33013)] and partially confirm the $$m=3$$ case of Conjecture 4.3 in the same paper. We also raise several related conjectures on $$q$$-congruences and supercongruences.

### MSC:

 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 11A07 Congruences; primitive roots; residue systems 11F33 Congruences for modular and $$p$$-adic modular forms

Zbl 1379.33013
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### References:

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