Related papers: Congruences for $q$-binomial coefficients
In this paper we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian Jacobi…
We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are…
This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let…
With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $\rho,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^\rho \cdot (-4)^{\rho(n-1-k)} \equiv 0\pmod{2^{\rho-2}n\binom{2n}{n}}.$$
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…
The q-binomial formula in the limit q->1 is shown to be equivalent to the Rogers five term dilogarithm identity.
We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.
The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the…
$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients…
Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom…
In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} ~(\bmod ~…
In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients…
The purpose of this paper is to give symmetric identities for higher-order degenerate q- Bernoulli polynomials arising from the p-adic q-integral on Zp.
Let $p$ be an odd prime. Define the Gaussian power sum \[ G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i]. \] We determine $G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$G_p(p)\equiv p^2(1+i)\pmod{p^3},$$…
In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1\pmod{4}$ or $a>1$ then $$…
In this paper we deduce some new supercongruences modulo powers of a prime $p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that $$\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…
We consider $cp_{a,b,m}(n)$, the number of $(a,b,m)$-copartitions of $n$. We find many infinitelymany congruencesmodulo 2 and 6 for some particular value of $a$, $b$ and $m$.
In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $\Phi_n(q)(1-aq^n)(a-q^n)$, where $\Phi_n(q)$ denotes the…
We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo…
Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues.