We give a new extension of Van Hamme's (A. 2) supercongruence with a parameter s by
establishing aq-analogue of this result. Our proof uses the 'creative microscoping'method …

Let $ p $ be an odd prime. In this paper, by using the properties of Legendre polynomials we
prove some congruences for $\sum _ {k= 0}^{\frac {p-1} 2}\binom {2k} k^{2} m^{-k}(\textrm …

For any positive integer n and variables a and x we define the generalized Legendre
polynomial P n (a, x) by P n (a, x)=∑ k= 0 n (ak)(− 1− ak)(1− x 2) k. Let p be an odd prime. In …

The Apéry polynomials are given by (Those An= An (1) are Apéry numbers.) Let p be an odd
prime. We show that and that for any p-adic integer x≢ 0 (modp). This enables us to …

Supercongruences involving Ap\'ery-like numbers and binomial coefficients Page 1 http://www.aimspress.com/journal/Math
AIMS Mathematics, 7(2): 2729–2781. DOI: 10.3934/math.2022153 Received: 21 July 2021 …

Let p> 3 be a prime, and let m be an integer with p∤ m. In the paper we solve some
conjectures of ZW Sun concerning∑ k= 0p− 1 (2kk) 3/mk (modp2),∑ k= 0p− 1 …

We collect here 100 open conjectures on congruences made by the author, some of which
have never been published. This is a new edition of the author's preprint arXiv: 0911.5665 …

Abstract Long and Ramakrishna (Adv. Math. 290: 773–808, 2016) gave the following
extension of Van Hamme's (H. 2) congruence: where p is an odd prime,(x) _k= x (x+ 1) ⋯ (x+ …

In this paper we deduce some new supercongruences modulo powers of a prime p> 3. Let
d∈{0, 1,…,(p− 1)/2}. We show that and where Ep− 3 (x) denotes the Euler polynomial of …

We consider the congruences modulo p 2 for the truncated hypergeometric functions. For
example, for any odd prime p and two p-integers α, β, we have the congruence …