Victor J. W. Guo

We give some new [Formula: see text]-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial, respectively. One of them is related to the following congruence due to Deines, Fuselier, Long, Swisher, and Tu: for primes [Formula: see text], [Formula: see text] where [Formula: see text] denotes the [Formula: see text]-adic Gamma function. Our proof relies on Watson’s [Formula: see text] transformation and the “creative microscoping” method introduced by the first author and Zudilin.

We prove that, for any odd prime [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] for [Formula: see text]. When [Formula: see text], this supercongruence reduces to a result of Guillera and Zudilin in 2012. We shall prove this result by building a [Formula: see text]-analogue of it. The main ingredients of the proof include the method of “creative microscoping” devised by Guo and Zudilin, a quadratic transformation of Rahman, a quadratic transformation of Gasper and Rahman, and Jackson’s [Formula: see text] summation. We also give a similar generalization of a supercongruence of Guo and Zudilin along with its [Formula: see text]-analogue.

By making use of a very-well-poised ϕ 5 6 {{}_{6}\phi_{5}} summation, the creative microscoping method, and the Chinese remainder theorem for coprime polynomials, we establish some new q -supercongruences, including some Dwork-type q -supercongruences. As a consequence, we obtain the following result: for any prime p ≡ 1 ( mod 4 ) {p\equiv 1\pmod{4}} and integer r ⩾ 1 {r\geqslant 1} , ∑ k = 0 p r - 1 2 ( 8 ⁢ k + 1 ) ⁢ ( 1 4 ) k 3 ⁢ ( 1 2 ) k k ! 3 ⁢ ( 3 4 ) k ≡ p r ( mod p r + 3 ) , \sum_{k=0}^{\frac{p^{r}-1}{2}}(8k+1)\frac{(\frac{1}{4})_{k}^{3}(\frac{1}{2})_{% k}}{k!^{3}(\frac{3}{4})_{k}}\equiv p^{r}\pmod{p^{r+3}}, where ( x ) 0 = 1 {(x)_{0}=1} and ( x ) n = x ⁢ ( x + 1 ) ⁢ ⋯ ⁢ ( x + n - 1 ) {(x)_{n}=x(x+1)\cdots(x+n-1)} for n ⩾ 1 {n\geqslant 1} .