In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.
In 2003, Rodriguez-Villegas conjectured four supercongruences on the truncated ${}_3F_2$ hypergeometric series for certain modular K3 surfaces, which were gradually proved by several authors. Motivated by some supercongruences on combinatorial numbers such as Apéry numbers and Domb numbers, we establish some new supercongruences on the truncated ${}_3F_2$ hypergeometric series, which extend the four Rodriguez-Villegas supercongruences.
We establish four supercongruences between truncated ${}_3F_2$ hypergeometric series involving $p$-adic Gamma functions, which extend some of the Rodriguez-Villegas supercongruences.
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and the other is regarded as $p$-adic analogue of an identity due to Ramanujan.
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured by Guo and Schlosser and recently confirmed by Jana and Kalita.
With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].
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 ${}_3F_2$ hypergeometric series. The corresponding conjectural supercongruences modulo $p^3$ are also proposed for further research.
We prove three more general supercongruences between truncated hypergeometric series and $p$-adic Gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for ${_7}F_6$ truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.
We prove two supercongruences for specific truncated hypergeometric series. These include an uniparametric extension of a supercongruence that was recently established by Long and Ramakrishna. Our proofs involve special instances of various hypergeometric identities including Whipple's transformation and the Karlsson--Minton summation.
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.
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which ar…
In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{4k}{2k+1}\binom{2k}k}{48^k}\equiv0\pmod{p^2},$$ and $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv1\pmod{3},\\ p/\binom{(2p+2)/3}{(p+1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv2\pmod{3}.\end{cases} $$ We also obtain some other results of such types.
In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$ \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{(1)_k^3}4^k\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}} $$ and $$ \sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac12)_k^4}{(1)_k^4}\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol and $B_0,B_1,B_2,\ldots$ are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. App…
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 certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Φ_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
We discuss two related principles for hypergeometric supercongrences, one related to accelerated convergence and the other to the vanishing of Hodge numbers. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.
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…
We give some new $q$-supercongruences on truncated forms of squares of basic hypergeometric series. Most of them are modulo the cube of a cyclotomic polynomial, and two of them are modulo the fourth power of a cyclotomic polynomial. The main ingredients of our proofs are the creative microscoping method, a lemma of El Bachraoui, and the Chinese remainder theorem for coprime polynomials. We also propose several related conjectures for further study.
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in e…
We prove supercongruences modulo $p^2$ for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are $d$ copies of $1/2$ and $d$ copies of $1$ for any integer $d\ge2$.
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n}\\ &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv -Γ_p\bigg(\frac{1}{n}\bigg)^n\pmod{p^3}, $$ where the truncated hypergeometric series $$_nF_{n-1} \bigg[\begin{matrix} x_1&x_2&\ldots&x_n\\ &y_1&\cdots&y_{n-1}\end{matrix}\bigg | \, z\bigg]_m=\sum_{k=0}^{m}\frac{z^k}{k!}\prod_{j=0}^{k-1}\frac{(x_1+j)\cdots(x_{n}+j)}{(y_1+j)\cdots(y_{n-1}+j)} $$ and $Γ_p$ denotes the $p$-adic gamma function. This confirms a c…
Using an intrinsic $q$-hypergeometric strategy, we generalise Dwork-type congruences $H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1})\pmod{p^3}$ for $s=1,2,\dots$ and $p$ a prime, when $H(N)$ are truncated hypergeometric sums corresponding to the periods of rigid Calabi--Yau threefolds.
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad \sum_{k=0}^{(p-1)/2}A_k\equiv a(p)\pmod{p^2}, $$ where $p>2$ is prime, $$ A_k=\prod_{j=0}^{k-1}\biggl(\frac{1/2+j}{1+j}\biggr)^3=\frac1{2^{6k}}{\binom{2k}k}^3 \quad\text{for}\ k=0,1,2,\dots, $$ and $a(p)$ is the $p$-th coefficient of (the weight 3 modular form) $q\prod_{j=1}^\infty(1-q^{4j})^6$. We complement our result with a general common $q$-congruence for related …
In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity $$ \sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2=16^n\sum_{k=0}^n\frac{\binom{n+k}{k}\binom{n}{k}\binom{2k}{k}^2}{(-16)^k} $$ which arises from a ${}_4F_3$ hypergeometric transformation. For any prime $p>3$, we prove that \begin{gather*} \sum_{n=0}^{p-1}\frac{n+1}{8^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2\equiv(-1)^{(p-1)/2}p+5p^3E_{p-3}\pmod{p^4},\\ \sum_{n=0}^{p-1}\frac{2n+1}{(-16)^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2\equiv(-1)^{(p-1)/2}p+3p^3E_{p-3}\pmod{p^4}, \end…
By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\…
In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.
Let $Φ_{n}(q)$ 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*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]Φ_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the…
