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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

Citations:

Zbl 1379.33013
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References:

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