Ling Long
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, we prove a conjecture of van Hamme.
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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.
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.
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 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 $d\ge2$.
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。
Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of F_p points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to supercongruences. This application uses two non-trivial generalized Harmonic sum identities discovered using the computer summation package Sigma. We illustrate the usage of Sigma in the discovery and proof …
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 prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are …
Two supercongruences involving truncated hypergeometric series
该研究利用超几何变换公式建立两个截断超几何级数的超同余式，分别证实第二作者的猜想、部分证实Guo等人的猜想并扩展Long和Ramakrishna的超同余式。
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.
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.
Using an identity due to Gessel and Stanton and some properties of the $p$-adic Gamma function, we establish a $p$-adic supercongruence for truncated hypergeometric series ${}_7F_6$. From it we deduce some related supercongruences, which extend certain recent results and confirm a supercongruence conjecture.
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 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.
Supercongruences arising from a ${}_7F_6$ hypergeometric transformation formula
本文借助 ${}_7F_6$ 超几何变换公式证明两个超同余式，其中之一证实 Guo–Liu–Schlosser 猜想并推广 Long–Ramakrishna 的结果。
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].
The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.
Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.
Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients
本文证明了三次对称超几何系数满足模 p⁴ 的超同余关系，通过 Ore 分解实现阶降，建立模算术字典并结合拉格朗日-布尔曼提取与三层指数截断，利用弗里克-海克 intertwining 方法消除缺陷形式。
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 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.
Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross-Koblitz formula. In this paper, we aim to prove four supercongruences between two truncated hypergeometric series ${}_2F_1$. This generalizes these four Rodriguez-Villegas supercongruences.
In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.
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.
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…
Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$ and supercongruences for hypergeometric polynomials
本文在模 $p^n$ 上实现 $\mathrm{GL}_2$ 的平凡表示，通过对称幂表示的直和子商构造，并归结为证明超几何多项式的新超同余。
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…
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;λ)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r λ^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of λ, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of ellipti…
We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo $p^k$ ($k>1$) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the $p$-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a trun…
$q$-Supercongruences Modulo the Third and Fifth Powers of a Cyclotomic Polynomial
该研究针对分圆多项式三次与五次幂模，建立基本超几何级数的q-超同余式，推广Van Hamme相关结果，采用创造性微缩放方法与Watson的₈φ₇变换公式完成证明。
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 supercongruences. We provide a framework of congruences covering all 22 cases. Using this framework we prove one of the outstanding supercongruence conjectures between a special value of a truncated ordinary hypergeometric series and the $p$-th Fourier coefficient of a modular form. In the course of this work we also establish two new binomial coefficient-harmon…
We give a simple unified proof for all existing rational hypergeometric Ramanujan identities for $1/π$, and give a complete survey (without proof) of several generalizations: rational hypergeometric identities for $1/π^c$, Taylor expansions, upside-down formulas, and supercongruences.
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.
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.
We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
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.
$p$-Adic hypergeometric functions and certain weight three newforms
本文通过证明经典超几何恒等式的 $p$-进类比，将某些权为三的新形式的第 $p$ 个傅里叶系数用 McCarthy 的 ${_3}G_3[\cdots]_p$ 特殊值表示，并由此重新证明了 Rodriguez-Villegas 猜想的超同余式。
In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating abelian squares, and we leave some remarks for the others.
Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$
本文利用 Wilf-Zeilberger 方法证明了孙智伟提出的两个关于截断超几何级数 ${}_4F_3$ 的超同余式，并给出其中一个涉及勒让德符号与欧拉数的模 $p^4$ 同余结果。
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.
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.
We prove some supercongruences for the truncated hypergeometric series.
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…
New convolution identities of hypergeometric Bernoulli polynomials are presented. Two different approaches to proving these identities are discussed, corresponding to the two equivalent definitions of hypergeometric Bernoulli polynomials as Appell sequences.
We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $ζ(3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence.
Weighted bilinear identities and supercongruences for Apéry-like polynomials
本文为两类Apéry型多项式建立加权双线性求和恒等式，结合端点值同余式得到模p^3和p^4的超同余式及特殊求值，并证实了Sun的一个猜想。
We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter $a$ when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.
We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.
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。
Supercongruences Arising from Truncated Appell Series
研究截断Appell级数和截断q - Appell级数的同余性质，利用组合恒等式和‘创造性显微镜’方法，给出截断Appell级数\(F_1\)和\(F_2\)一些同余式的q - 类似，建立与\(F_4\)相关新q - 超同余式并猜想。
We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special values of that function.
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for $1/π$. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Section \ref{section-pi2} are of the same style but related to Ramanujan-like series for $1/π^2$.
A full $p^{4r}$ supercongruence tower for a level-three symmetric-cube hypergeometric sequence
本文证明了一个三级对称立方超几何序列的素数幂超同余塔 $A_{mp^r}\equiv A_{mp^{r-1}}\pmod{p^{4r}}$，通过模形式方法将已知的深度一模 $p^4$ 推广到所有素数幂。
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
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\…
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