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q -Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q -supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ϕ 7 8 summation, Watson's ϕ 7 8 transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a q -analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773–808] and two q -supercongruences involving double series. [ABSTRACT FROM AUTHOR]

In terms of the derivative operator, integral operator and a Saalschütz-type identity, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

In terms of a q -analogue of Whipple's F 2 3 summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish a q -supercongruence modulo the fifth power of a cyclotomic polynomial. More concretely, we give a q -analogue of a result by Swisher and a q -analogue of a conjecture by Liu. [ABSTRACT FROM AUTHOR]

In terms of the derivative operator and Baily's F 1 2 ( 1 2 ) -series identity, two kinds of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

In 2017, He (2017) [9] established two supercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu (2017) [11] 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. [ABSTRACT FROM AUTHOR]

Abstract: In terms of the hypergeometric method, we establish the extensions of two formulas for due to Ramanujan (1914) [32]. Further, other five summation formulas for with free parameters are also derived in the same way. [Copyright &y& Elsevier]

Abstract: In terms of the hypergeometric method, we establish ten general -formulas with free parameters which include several known results as special cases. [Copyright &y& Elsevier]

Abstract: By applying the triple form of Gould–Hsu inversions to the Pfaff–Saalschütz theorem, a family of summation formulas on the Fox–Wright function, which implies several known -series identities due to Chen and Chu (2011) and numerous new hypergeometric series identities, is established. [Copyright &y& Elsevier]

Abstract: Paule and Schneider (2003) , and Chu (Chu and Donno) (2005) gave a family of wonderful harmonic number identities. Their generalized versions associated with q-harmonic numbers will be established by applying a derivative operator to Watson''s q-Whipple transformation. [Copyright &y& Elsevier]

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q -supercongruence with two parameters modulo [ n ] Φ n (q) 3. Here [ n ] = (1 − q n) / (1 − q) and Φ n (q) is the n -th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q -analogue of Long's supercongruence. The latter is also a generalization of a recent q -supercongruence obtained by Guo and Schlosser. [ABSTRACT FROM AUTHOR]