In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic number.
By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'Hôpital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercon…
In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.
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…
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials.
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 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.
In this paper we deduce some new supercongruences modulo powers of a prime $p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{8^k}\equiv 0\ (\mbox{mod}\ p)\ \ \ \mbox{if}\ d\equiv \frac{p+1}2\ (\mbox{mod}\ 2),$$ and $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{16^k} \equiv\left(\frac{-1}p\right)+p^2\frac{(-1)^d}4E_{p-3}\left(d+\frac12\right)\pmod{p^3},$$ where $E_{p-3}(x)$ denotes the Euler polynomial of degree $p-3$, and $(-)$ stands for the Legendre symbol. The paper also contains some other results such as $$\sum_{k=0}^{p…
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we prove some supercongruences concerning $$\align &\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k}{54^k},\ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{4k}{2k}}{128^k},\ \sum_{k=0}^{p-1}\frac{\binom{3k}k\binom{6k}{3k}}{432^k}, &\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{3k}{k}}{m^k}, \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k},\ \sum_{k=0}^{p-1}\f{\binom{2k}k\binom{3k}{k}\binom{6k}{3k}}{m^k}\mod {p^2}.\endalign$$ Thus we solve some conjectures of Zhi-Wei Sun and the author.
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Φ_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
In this paper, a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools are Watson's $_8ϕ_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials. By taking suitable parameter substitutions in the established $q$-supercongruence, some nice congruences involving the Bernoulli numbers are derived.
In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.
Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and $\sum_{k=0}^{p-1} \binom ak^r(1-\frac 2ak)^s$ modulo $p^4$, where $r\in\{3,4\}$ and $s\in\{1,3\}$.
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 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.
We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.
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…
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}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ \sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2 }{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.
Let $Φ_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $Φ_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this creative microscoping method to confirm some conjectures on $q$-supercongruences modulo $Φ_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas: $$ \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k}…
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 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.
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 $d\ge2$.
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.
We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $α$ is $p$-integral, then we can completely determine $$ \sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k}\quad\text{and}\quad \sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k} $$ modulo $p^3$. In particular, by setting $α=-1/2$, we confirm two conjectured congruences of Z.-W. Sun.
In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom nk(-1)^k\binom xk\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}(\pm1)^kd_k(x)^2$ and $\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\equiv(-1)^{\langle x\rangle_p}\pmod{p^2},$$ where $\langle x\rangle_p$ denotes the least nonnegativ…
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 generalizations of the harmonic numbers. We apply the technique to derive many new supercongruences. We also provide software for finding and proving supercongruences using our technique.
Using Watson's terminating $_8ϕ_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.
Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ is not divisible by $p$. We prove that if a sufficiently regular $n$-variable function is zero modulo $p^r$ over some suitable collection of $r$ hyperplanes, then it is zero modulo $p^r$ over the whole ${\mathbb Z}_p^n$. We provide various applications of this general criterion by establishing several $p$-adic analogues of hypergeometric identities.
In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number of nonzero binomial coefficients modulo p^alpha as a sum over certain integer partitions. For fixed alpha, this expression can be rewritten to show explicit dependence on the number of occurrences of each subword in the base-p representation of n.
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.
For each of the $15$ known sporadic Apéry-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of $ζ(3)$. Moreover, we show that each of these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.
The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv 0\pmod{n{2n\choose n}}, \\ \sum_{k=0}^{p-1}\frac{3k+1}{(-16)^k} {2k\choose k} f_k &\equiv p (-1)^{\frac{p-1}{2}} \pmod{p^3}. \end{align*} where $n>1$ and $p$ is an odd prime. The second congruence modulo $p^2$ confirms a recent conjecture of Z.-H. Sun. We also show that, if $p$ is a prime of the form $4k+3$, then $$ \sum_{k=0}^{p-1}\frac{{2k\choose k} f_k}{(-16)^k} \equiv 0 …
In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$.
For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\frac{1}{64^k} \equiv 0 \pmod{p^2} \quad\text{for}\quad p\equiv 3\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^…
In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, i.e. at `supercongruences', we find interesting relations linking them together, as well as linking them with arithmetic derivatives and zeta values. In the current work, we expand on the first analog and connections with arithmetic derivatives more systematically, giving many more equivalent conditions linking the two, now using `mixed derivati…
Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of F_p points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to supercongruences. This application uses two non-trivial generalized Harmonic sum identities discovered using the computer summation package Sigma. We illustrate the usage of Sigma in the discovery and proof …
The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences: $$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\ \mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\ \mbox{modulo}\ p$$ for any prime $p>3$, the second one was conjectured by Z.-W. Sun in 2012. These two congruences are very important to prove the following conjectures of Z.W.Sun: For any old prime $p$, we have $$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\bi…
We show that for any mod $2^m$ characters, $χ_1, χ_2,$ the complete exponential sum, $$ \sum_{x=1}^{2^m}χ_1(x) χ_2(Ax^k+B), $$ has a simple explicit evaluation.
Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}/m^k$ mod p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show that if h\not=0 (mod p) and p^a>3 then $$ sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}(-h/2)^k =(\frac{1-2h}{p^a})(1+h((4h-2)^{p-1}/h^{p-1}-1)) (mod p^2),$$ where (-) denotes the Jacobi symbol. Here is another remarkable congruence: If p>3 then $$\sum_{k=0}^{p^a-1}\binom{p^a-1}{k}\binom{2k}{k}(-1)^k =3^{p-1}(\frac{p^a}3) (mod p^2).$$
We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).
I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.
For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.
In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Apéry-like numbers modulo $p^3$, where $p$ is an odd prime.
