A congruence involving products of $q$-binomial coefficients

A generalization of the Gaussian formula and a q-analog of Fleck's congruence

The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n of Pascal's triangle is captured by the so-called Gaussian Formula. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients.

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.

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…

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.

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period $ν_m(q)$ such that for any integers n>0 and r we have $[n+ν_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.

Three pairs of congruences concerning sums of central binomial coefficients

Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.

On some new congruences for binomial coefficients

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\sum_{k=0}^{p^a-1}\binom{2k}{k+δ}$ mod $p^3$ for $δ=0,1$. We also show that $$C_n^{-1}\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol.

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.

Proof of Sun's conjectures on super congruences and the divisibility of certain binomial sums

In this paper, we prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv p+2\left(\frac{-1}{p}\right)p^3E_{p-3}\pmod{p^4},$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers.

Congruences for $q$-binomial coefficients

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial ratios.

Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers $h \geq 2$

We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{x+n}{n}$ and $\binom{x}{n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \geq 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series gener…

Congruences concerning binomial coefficients and binary quadratic forms

Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{(-144)^k}\ \text{and} \ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{648^k}$$ modulo $p^2$, and partial results for $\sum_{k=0}^{(p-1)/2} \binom{2k}k^3\frac{w(k)}{m^k}$ modulo $p^2$, where $m\in\{1,16,-64,256,-512,4096\}$ and $w(k)\in\{k^2,k^3,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3}, \frac 1{2k-1},\frac 1{k+2}\}$.

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…

On congruences related to central binomial coefficients

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=π/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=π/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/2<k<p}\binom{2k}{k}/((2k+1)4^k)=3\sum_{p/2<k<p}\binom{2k}{k}/((2k+1)16^k)= pE_{p-3} (mod p^2),$$ where p>3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=…

Some congruences for the second-order Catalan numbers

Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.

New congruences for central binomial coefficients

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.

An inductive proof of Straub's q-analogue of Ljunggren's congruence

Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.

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.

Proof of a congruence on sums of powers of $q$-binomial coefficients

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where $[n]=\frac{1-q^n}{1-q}$, $[n]!=[n][n-1]\cdots[1]$, and ${a\brack b}=\prod_{k=1}^b\frac{1-q^{a-k+1}}{1-q^k}$. The $a_1=\cdots=a_m$ case confirms a recent conjecture of Z.-W. Sun. We also show that, if $p>\max\{a,b\}$ is a prime, then \begin{align*} \frac{[a+b+1]!}{[a]![b]!}\sum_{h=0}^{p-1}q^h{h\brack a}{h\brack b} \equiv (-1)^{a-b} q^{ab-{a\choose 2}-{b\choose 2}}[p…

素数基二项式同余的结构化解

Structured Solutions of Prime-Base Binomial Congruences

研究素数基 q 下二项式同余式 C(qn, n) ≡ q^n (mod n) 的解结构，将解 n = q^t p 转化为模 p 同余和数字和不等式两个独立条件，并用于生成新解及证明平方解与 Wieferich 素数等价。

New conjectures involving binomial coefficients and Apéry-like numbers

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.

Lucas定理的一个推广

An extension of Lucas Theorem

本文给出计算模素数二项式系数的同余判据的初等证明，并据此推广Lucas定理，应用于推导二项式系数和的恒等式与同余式。

Decompositions of commutative monoid congruences and binomial ideals

Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground fiel…

Binomial coefficient-harmonic sum identities associated to supercongruences

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…

On the divisibility of sums of even powers of $q$-binomial coefficients

We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.

On sums related to central binomial and trinomial coefficients

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic forms, and 62 proposed new series for $1/π$ motivated by congruences and related dualities.

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…

On b-ary binomial coefficients with negative entries

Abstract. We generalize the $b$-ary binomial coefficients with negative entries, which is based on the generating function obtained in early work. Besides an explicit expression involving the restricted partition, several properties such as symmetry, congruence and Pascal-like recurrence are studies. Finally, we also provide two different generalizations, partially satisfying Pascal-like recurrences.

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}…

Divisibility of some binomial sums

With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $ρ,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^ρ\cdot (-4)^{ρ(n-1-k)} \equiv 0\pmod{2^{ρ-2}n\binom{2n}{n}}.$$

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in several contexts.

New congruences involving Apéry-like numbers

In this paper we present many congruences for several Apéry-like sequences.

On two congruences involving Apéry and Franel numbers

In this paper, we generalise a congruence which proved by V. J.-W. Guo and J. Zeng \cite{gz-jnt-2012} involving Apéry numbers, and we obtain a congruence involving Franel numbers which confirms a congruence conjecture of Z.-W. Sun \cite[Conjecture 57(ii)]{sun-njdx-2019}.

On a congruence involving $q$-Catalan numbers

Based on a $q$-congruence of the author and Petrov, we set up a $q$-analogue of Sun--Tauraso's congruence for sums of Catalan numbers, which extends a $q$-congruence due to Tauraso.

On the divisibility of $q$-trinomial coefficients

We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.

Some new results about $q$-trinomial coefficients

In this paper, we present several new congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. A new congruence on sums of central $q$-binomial coefficients is also established.

Two congruences involving Andrews-Paule's broken 3-diamond partitions and 5-diamond partitions

In this note, we will give proofs of two congruences involving broken 3-diamond partitions and broken 5-diamond partitions which were conjectured by Peter Paule and Silviu Radu.

Some Elementary Congruences for the Number of Weighted Integer Compositions

An integer composition of a nonnegative integer $n$ is a tuple $(π_1,\ldots,π_k)$ of nonnegative integers whose sum is $n$; the $π_i$'s are called the parts of the composition. For fixed number $k$ of parts, the number of $f$-weighted integer compositions (also called $f$-colored integer compositions in the literature), in which each part size $s$ may occur in $f(s)$ different colors, is given by the extended binomial coefficient $\binom{k}{n}_{f}$. We derive several congruence properties for $\binom{k}{n}_{f}$, most of which are analogous to those for ordinary binomial coefficients. Among the…

q-Analogs of some congruences involving Catalan numbers

We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.

Proof of some divisibility results on sums involving binomial coefficients

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.

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.

Curious congruences for Fibonacci numbers

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\not=2,5$ is a prime then $$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and $$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.

Congruences arising from Apéry-type series for zeta values

Recently, R. Tauraso established finite $p$-analogues of famous Apéry series for $ζ(2)$ and $ζ(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Apéry-type series for $ζ(4)$ and $ζ(5).$ We also prove a $p$-analogue of Zeilberger's series for $ζ(2)$ confirming a conjecture of Z. W. Sun.

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.

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.

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。

关于类Apéry序列的二项式变换

On the binomial transforms of Apéry-like sequences

本文研究了15个类Apéry序列在$u_n\equiv α^n \pmod{N_α}$形式下的同余问题，通过二项式变换这一统一方法确定了所有序列的$\max\{N_α|α\in \mathbb{Z}\}$值。

Lucas-type congruences for cyclotomic $ψ$-coefficients

Let p be any prime and a be a positive integer. For nonnegative integers l,n and an integer r, the normalized cyclotomic $ψ$-coefficient $${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod p^a)}(-1)^k{n \choose k}{{(k-r)/p^a} \choose l}$$ is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.

三个涉及中心二项式系数的组合和

Three combinatorial sums involving central binomial coefficients

本文研究三类含中心二项式系数及不同调和数的组合和，通过分部求和建立递归式，并用第二类 Stirling 数与 r-Stirling 数给出闭合形式。

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 $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.

Algebraic independence of $G$-functions and congruences "à la Lucas"

We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the $G$-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence…

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 $p^2$.