Some q-supercongruences modulo the square and cube of a cyclotomic polynomial

Guo, Victor J. W.; Schlosser, Michael J.

doi:10.1007/s13398-021-01070-y

Cited by 5 publications

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“…Further, the present authors [ 8 ] proved the following q -supercongruence: Let d and r be odd integers satisfying , (in particular, r may be negative) and . Let n be an integer such that and .…”

Section: Introductionmentioning

confidence: 90%

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Three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial

Guo

Schlosser

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper, three parametric q-supercongruences for truncated very-well-poised basic hypergeometric series are proved, one of them modulo the square, the other two modulo the cube of a cyclotomic polynomial. The main ingredients of proof include a basic hypergeometric summation by George Gasper, the method of creative microscoping (a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

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Section: Introductionmentioning

confidence: 90%

“…Let n be an integer such that and . Then Note that in [ 8 , Conjecture 3] it was even conjectured that the q -supercongruence ( 1.7 ) is true modulo for , which still remains open.…”

Section: Introductionmentioning

confidence: 99%

Three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial

Guo

Schlosser

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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New q-Analogues of Van Hamme’s (E.2) Supercongruence and of a Supercongruence by Swisher

Guo

Schlosser

2023

Results Math

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In this paper, a couple of q-supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new q-analogue of the (E.2) supercongruence by Van Hamme, another one is a new q-analogue of a supercongruence by Swisher, while the other results are closely related q-supercongruences. The proofs make use of special cases of a very-well-poised $${}_6\phi _5$$ 6 ϕ 5 summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

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