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Results in Mathematics

, 74:131 | Cite as

Common q-Analogues of Some Different Supercongruences

  • Victor J. W. GuoEmail author
Article

Abstract

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 Guo (Integral Transform Spec Funct 28:888–899, 2017) 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.

Keywords

Basic hypergeometric series Watson’s transformation q-congruences supercongruences cyclotomic polynomial 

Mathematics Subject Classification

33D15 Secondary 11A07 11F33 

Notes

Acknowledgements

The author thanks the anonymous referee for very useful comments that helped to improve the quality of the article. Thanks also to Michael J. Schlosser for a careful reading of the introduction. This work was partially supported by the National Natural Science Foundation of China (Grant 11771175).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaiyin Normal UniversityHuai’anPeople’s Republic of China

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