Supercongruences for Almkvist--Zudilin sequences

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

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 Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.

A $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] developed an analytical method, called `creative microscoping', to prove many supercongruences by establishing their $q$-analogues. In this paper, we apply this method to give a $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences, which was recently conjectured by Guo and Zudilin.

q-Analogues of two "divergent" Ramanujan-type supercongruences

Guillera and Zudilin proved three "divergent" Ramanujan-type supercongruences by means of the Wilf-Zeilberger algorithmic technique. In this paper, we prove $q$-analogues of two of them via the $q$-WZ method. Additionally, we give $q$-analogues of two related congruence of Sun, one is confirmed and the other is conjectural.

Multivariate Apéry numbers and supercongruences of rational functions

One of the many remarkable properties of the Apéry numbers $A (n)$, introduced in Apéry's proof of the irrationality of $ζ(3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes $p \geq 5$. Similar congruences are conjectured to hold for all Apéry-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Apéry numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \begin{equation*} \frac{1}{(1 - x_1 - x_2) (1 - x_3…

On Some Hypergeometric Supercongruence Conjectures of Long

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version…

Van Hamme的G.2超同余的q-模拟

q-analogues of the G.2 Supercongruence of Van Hamme

该研究受同余与q-同余研究启发，用创造性显微术方法首次给出Van Hamme的（G.2）超同余的两种直接q-模拟，还提出了与Swisher的Dwork型超同余（G.3）相关的猜想。

Two $q$-supercongruences from Watson's transformation

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] introduced a new method called `creative microscoping', to prove many $q$-supercongruences in a unified way. In this paper, we apply this method and Watson's ${}_8ϕ_7$ transformation formula to prove two $q$-supercongruences, which were recently conjectured by Guo and Schlosser.

Van Hamme的(E.2)与(F.2)超同余式及Swisher的两个超同余式的改进

Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher

该研究利用更一般的WZ对，将Van Hamme的(E.2)、(F.2)超同余式及Swisher的两个超同余式推广到奇素数$p$的模$p^4$情形，且推广结果与Euler多项式相关，同时提出了相关$q$-同余猜想供后续研究。

Bivariate identities for values of the Hurwitz zeta function and supercongruences

In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.

Supercongruences for sums involving Domb numbers

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime $p\ge 5$, \begin{align*} \sum_{k=0}^{p-1}\frac{3k+1}{(-32)^k}{\rm Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3} \pmod{p^4}, \end{align*} which is regarded as a $p$-adic analogue of the following interesting formula for $1/π$ due to Rogers: \begin{align*} \sum_{k=0}^{\infty}\frac{3k+1}{(-32)^k}{\rm Domb}(k)=\frac{2}π. \end{align*} Here ${\rm Domb}(n)$…

A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial

In this paper, a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools are Watson's $_8ϕ_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials. By taking suitable parameter substitutions in the established $q$-supercongruence, some nice congruences involving the Bernoulli numbers are derived.

On $p$-adic expansions of Ramanujan-like series

Inspired by a Zudilin-Zhao's supercongruences pattern related to Ramanujan-like series for $1/π^k$, we conjecture a kind of $p$-adic expansions.

两个超几何超同余中的变形与对称性

Deformation and Symmetry in Two Hypergeometric Supercongruences

该研究证明了Li-Long-Tu的两个超同余猜想，通过变形方法结合超几何变换与$W(D_5)$对称性归约问题，为超同余提供统一解释并拓展了方法适用范围。