Related papers: Supercongruences arising from a ${}_7F_6$ hypergeo…

For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We…

We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of q-binomial theorem and a…

Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of…

In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the…

In this article, we list a few hypergeometric supercongruence conjectures based on two evaluation formulas of Whipple and numeric data computed using Magma and Sagemath.

In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified 22 possible…

In 2021, the first author and Kalita obtained two general hypergeometric formulas for sums involving certain rising factorials to prove some supercongruence conjectures of Guo related to (B.2) and (C.2). In this paper, we further generalize…

We establish some supercongruences for the truncated ${}_2F_1$ and ${}_3F_2$ hypergeometric series involving the $p$-adic Gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated ${}_3F_2$…

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…

A supercongruence is a congruence between rational numbers modulo a power of a prime. In this paper, we give a technique for finding and algorithmically proving supercongruences by expressing terms as infinite series involving certain…

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*}…

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These…

Inspired by the recent work of Guo, we establish some new q-supercongruences including q-analogues of some Ramannujan-type supercongruences, by using the Bailey transformation formula and the `creative microscoping' method recently…

In this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular,…

Some multiple hypergeometric transformation formulas arising from the balanced du- ality transformation formula are discussed through the symmetry. Derivations of some transformation formulas with different dimensions are given by taking…

An elementary proof is given for a nonterminating "strange" cubic $_7F_6$-series summation formula of Gasper and Rahman, through the modified Abel lemma on summation by parts. As a byproduct, an interesting nonterminating…

In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22…

We extend the validity range of a Ramanujan's hypergeometric transformation formula proved by Berndt, Bhargava and Garvan, Trans. Amer. Math. Soc. 347, 4163 (1995) and study its implications. Relations to special values of complete elliptic…

Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…