New congruences involving Apéry-like numbers

A simple proof of a congruence for a series involving the little $q$-Jacobi polynomials

We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

On two congruences involving Franel numbers

Via symbolic summation method, we establish the following series for $π^2$: \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} = \frac{π^2}{18}, \end{align*} where $H_k=\sum_{j=1}^k 1/j$. We also derive a $p$-adic congruence related to this series. As an application, we prove two congruences involving Franel numbers, one of which was originally conjectured by Sun.

涉及正交多项式和Apéry类数的恒等式与同余式

Identities and congruences involving orthogonal polynomials and Apéry-like numbers

本文建立三项递推序列的一般恒等式，应用于正交多项式和Apéry类数，给出相关恒等式与同余式。

Congruences involving binomial coefficients and Apéry-like numbers

For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}^{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3)^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Apéry-like sequence. In this paper we deduce many congruences involving $\{W_n\}$, in particular we determine $\sum_{k=0}^{p-1}\binom{2k}k\frac{W_k}{m^k}\pmod p$ for $m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243$ by using binary quadratic forms, where $p>3$ is a prime. We also prove several congruences for generalized Apéry-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients an…

Congruences involving binomial coefficients and Lucas sequences

In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p>5 is a prime then $\sum_{k=0}^{p-1}F_k*binom(2k,k)/12^k$ is congruent to 0,1,-1 modulo p according as p=1,4 (mod 5), p=13,17 (mod 30), and p=7,23 (mod 30) respectively, where {F_n} is the Fibonacci sequence. We also raise several conjectures.

New super congruences involving Bernoulli and Euler polynomials

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k \quad\text{and}\quad\sum_{k=0}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}{2k-1}\pmod {p^2}$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.

Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.

Congruences involving $\binom{4k}{2k}$ and $\binom{3k}k$

Let $p$ be a prime greater than 3. In the paper we mainly determine $\sum_{k=0}^{[p/4]}\binom{4k}{2k}(-1)^k$, $\sum_{k=0}^{[p/3]}\binom{3k}k, \sum_{k=0}^{[p/3]}\binom{3k}k(-1)^k$ and $\sum_{k=0}^{[p/3]}\binom{3k}k(-3)^k$ modulo $p$, where $[x]$ is the greatest integer not exceeding $x$.

A congruence concerning a convolution involving weighted Bernoulli numbers

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.

Congruences involving $\binom{2k}k^2\binom{3k}km^{-k}$

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{3k}km^{-k}\mod {p^2}$.

A remarkable congruence involving Appell polynomials

In this paper, by the umbral calculus method, we give a remarkable congruence involving Appell polynomials. Some applications on derangement polynomials are also presented.

Conjectures on congruences involving binomial coefficients and Apéry-like numbers

In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Apéry-like numbers.

Congruences for sums involving products of three binomial coefficients

Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3},\frac 1{2k-1},\frac 1{k+2}, \frac 1{k+3}, k,k^2,k^3,\frac 1{a+k},\frac 1{a+k-1}.$$ As consequences, taking $a=-\frac 12,-\frac 13,-\frac 14,-\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.

Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$

Define $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ for $n=0,1,2,...$. Those numbers $g_n=g_n(1)$ are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\sum_{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\sum_{k=1}^{p-1}\frac1k\equiv0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{1}{k^2}\equiv0\pmod p$$ for any prime $p>3$.

Some congruences involving binomial coefficients

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \left(\frac p3\right)(3^{p-1}-1)\ \pmod{p^3}.$$ In addition, we get some new combinatorial identities.

A congruence involving the quotients of Euler and its applications (III)

In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).

Super-congruences involving trininomial coefficients

The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of the polynomial $1+x+x^{2}.$ In main results we extend some known congruences involving the binomial coefficients $\binom{np-1}{p-1}$ and $\binom{np-1}{\left( p-1\right) /2}$ and establish congruences link binomial coefficients and harmonic numbers.

关于涉及Apéry数的同余

On congruences involving Apéry numbers

本文证明了Z.-W. Sun猜想的一个涉及Apéry数求和的同余式，对任意素数p>3和正奇数m，存在仅依赖m的p-adic整数c_m使该同余式成立。

Some parametric congruences involving generalized central trinomial coefficients

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}}{12^k}T_k\equiv\left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{T_kH_k}{3^k}\equiv\frac{3+\left(\fr…

Congruences involving Franel and Catalan-Larcombe-French numbers

Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let $S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$. We also determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$ for $m=7,16,25,32,64,160,800,1600, 156832$.

Congruences involving the reciprocals of central binomial coefficients

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

Identities and congruences involving the Fubini polynomials

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_{x}))^{p}\equiv f(F_{x})$ and we give other curious congruences.

Some $q$-analogs of congruences for central binomial sums

We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.

Congruences for two types of Apery-like sequences

In this paper we present many results and conjectures on congruences involving two types of Apéry-like sequences $\{G_n(x)\}$ and $\{V_n(x)\}$.

Congruence identities of regularized multiple zeta values involving a pair of index sets

Riemann zeta values are generalized to multiple zeta values (MZVs) by use of nested sums, and MZVs are generalized to regularized multiple zeta values (RMZVs) by regularization of divergent infinite series. In the present paper, we prove congruence identities of RMZVs of depth $n$ involving a pair of index sets; the congruence relation is given by the vector space spanned by MZVs of depth $n-1$ and products of MZVs. We also obtain a proof of the parity result, and a congruence sum formula for MZVs.

Congruences on the Bell polynomials and the derangement polynomials

In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.

克里斯托费尔-达布公式的推广及类阿佩里数相关同余

Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers

本文将正交多项式的克里斯托费尔-达布公式推广到一般三项递推序列，研究两类特定递推序列$g_n(x)$、$v_n(x)$的恒等式与同余关系，推进了相关数论问题的研究。

$q$-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter's solution of the hard-hexagon model

For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. For each $s\in\{1,3\}$, the generating function of $R^*_s(n)$, the number of partitions of $n$ into parts not congruent to $0$, $\pm s\pmod {10}$ and $10-2s \pmod {20}$, arises naturally in regime IV of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $R_s(n)$ and $R^*…

涉及二项式系数和费马商的某些同余式

Some Congruences Involving Binomial Coefficient and Fermat Quotient

本文研究了涉及二项式系数和费马商的某些同余式，探讨了相关求和形式的性质。

涉及格点路径与整数分拆的双射及同余关系

Bijections and congruences involving lattice paths and integer compositions

该研究通过建立不同变体Dyck路径与整数分拆的新双射，解释计数公式4^{n-1}的组合意义，推导路径同余与分拆奇偶性结果，还揭示其与多个组合数学对象的关联，深化了相关结构认知。

Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{Φ_n(q)}, $$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $Φ_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, $$ for $a,m\ge…

New congruences for sums involving Apery numbers or central Delannoy numbers

The Apéry numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\sum_{k=0}^{n}{n+k\choose 2k}^2{2k\choose k}^2, \quad D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences: \sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\equiv \sum_{k=0}^{n-1}\varepsilon^k (2k+1)^{2r+1}D_k \equiv 0\pmod n, where $n\geqslant 1$, $r\geqslant 0$, and $\varepsilon=\pm1$. For $r=1$, we further show that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \quad \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where $p>3$ is a prime. …

Congruences involving alternating multiple harmonic sum

We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.

孙智宏提出的若干涉及类Apéry数的同余猜想的证明

Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers

该研究证明了孙智宏提出的一个涉及类Apéry数的素数模p⁴同余猜想，明确了含Franel数的求和式与Euler数的同余关系。

Factors of certain sums involving central q-binomial coefficients

Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pan's technique and a simple $q$-congruence observed by Guo and Schlosser.

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 Curious Congruence Involving Alternating Harmonic Sums

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{{{(-1)}^{i}}}{ijk}}\equiv \frac{{{p}^{r-1}}}{2}{{B}_{p-3}}\, (\bmod \, {{p}^{r}}).\]

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $χ_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)=\sum_{x=1}^{[n/m]}\frac{χ_n(x)}{x^k}\left(\bmod n^{r+1}\right)$ for $r\in \{1,2\}$, any positive integer $m $ with $n \equiv \pm 1 \left(\bmod m \right)$ in terms of Bernoulli polynomials. As its an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers.

Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…

Two congruences involving harmonic numbers with applications

The harmonic numbers $H_n=\sum_{0<k\le n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}kH_k\equiv\frac13\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod{p}$$ and $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}kH_{2k}\equiv\frac7{12}\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine $\sum_{n=1}^{p-1}g_n$ and $\sum_{n=1}^{p-1}h_n$ modulo…

Congruences involving generalized central trinomial coefficients

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\sum_{k=0}^n\binom nk\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\ldots$ we have $$\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\equiv0\pmod{n^2}$$ and in particular $n^2\mi…

Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.

Some $q$-congruences involving central $q$-binomial coefficients

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for all $d=0,1, \ldots n$ and $n= p^a$, where $a$ is a positive integer. In this paper, we present some $q$-analogues of these congruences in the cases $m=2, 4$ for any positive integer $n$.

Some congruences involving powers of Delannoy polynomials

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3 &\equiv p\left(\frac{-4x-3}{p}\right) \pmod{p^2}, \\ \sum_{k=0}^{p-1}(2k+1)D_k(x)^4 &\equiv p \pmod{p^2}, \\ \sum_{k=0}^{p-1}(-1)^k(2k+1)D_k(x)^3 &\equiv p\left(\frac{4x+1}{p}\right) \pmod{p^2}, \end{align*} where $\big(\frac{\cdot}{p}\big)$ denotes the Legendre symbol. The first two congruences confirm a conjecture of Z.-W. Sun [Sci. China 57 (2014), 1375--1400].…

Congruences of multiple sums involving invariant sequences under binomial transform

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd} -{2^{n+1}+4\over n6^n} B_{p-n}({1\over 3}) &\pmod{p} &{if $n$ is even}. $$ where $n$ is a positive integer and $p$ is prime such that $p>\max(n+1,3)$.

$6$-regular partitions: new combinatorial properties, congruences, and linear inequalities

We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-reg…

Cubic congruences and sums involving $\binom{3k}k$

Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum $\sum_{k=1}^{[p/3]}\binom{3k}ka^k$, where $[x]$ is the greatest integer not exceeding $x$. Suppose that $a_1,a_2,a_3$ are rational p-adic integers, $P=-2a_1^3+9a_1a_2-27a_3$, $Q=(a_1^2-3a_2)^3$ and $PQ(P^2-Q)(P^2-3Q)(P^2-4Q)\not\equiv 0\pmod p$. In this paper we show that the number of solutions of the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$ depends only on $\sum_{k=1}^{[p…

Proof of some conjectural congruences involving Domb numbers

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, $$ and if $p\equiv2\pmod3$, then $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\frac{p-5}6}^{-2} \pmod{p^3}, $$ where $D_n=\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}$ stands for the $n$th Domb number.

A New Super Congruence Involving Multiple Harmonic Sums

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{5}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{5}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\equiv -\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \pmod{{{p}^{r}}}. \end{equation} This gives an extension of a family of super congruences found by Wang, Cai and Zhao.

Various congruences involving binomial coefficients and higher-order Catalan numbers

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study congruences involving higher-order Catalan numbers $C_k^{(h)}=\binom[(h+1)k,k]/(hk+1)$ and $\bar C_k^{(h)}=\binomal[(h+1)k,k]*h/(k+1)$. Our tools include linear recurrences and the theory of cubic residues. Here are some typical results in the paper. (i) If $p^a=1 (mod 6)$ then $$\sum_{k=1}^{p^a-1}\binom[3k,k]/6^k=2^{(p^a-1)/3}-1 (mod p).$$ Also, $$\sum_{k=0}^{p^a-1}…

关于涉及$\binom{rk}{k}$的和的同余式

Congruences for sums involving $\binom{rk}{k}$

研究了模$p$下形式为$\sum_k\binom{rk}{k}x^k/k$的有限和的同余性质，通过两种互补方法获得模$p^2$下的同余式，用有限多项式对数表达式表示。

New congruences involving products of two binomial coefficients

Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv1\pmod 4$ or $a>1$ then $$\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\frac{\binom{2k}k^2}{16^k}\equivł(\frac{-1}{p^a}\r)\pmod{p^3}$$ with $(-)$ the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences: \begin{align*}\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{3k}k}{27^k}\equiv&ł(\frac p3\r)\frac{2^p+1}3\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{6k}{3k}\binom{3k}k}{(2k+1)432^k}\equiv&ł(\frac p3\r)\frac{3^p+1}4\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{4k}{2k}\bin…

Telescoping method and congruences for double sums

In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.

Supercongruences on some binomial sums involving Lucas sequences

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.