$q$-Supercongruences from Gasper and Rahman's summation formula

DOI10.1016/J.AAM.2022.102376arXiv2111.07595WikidataQ113881573 ScholiaQ113881573MaRDI QIDQ6383053

Publication date: 15 November 2021

Abstract: In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several

q

-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for basic hypergeometric series. As conclusions, He's first conjecture is confirmed and a more general form of He's second conjecture is proved.

Mathematics Subject Classification ID

Binomial coefficients; factorials; (q)-identities (11B65) Basic hypergeometric functions in one variable, ({}_rphi_s) (33D15) Congruences; primitive roots; residue systems (11A07)

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