Related papers: Congruences for $q$-binomial coefficients

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by…

Andrews once gave $q$-analogues of a binomial congruence of Glaisher, and he suggested perfect $q$-analogues. In this note we give ones meeting the demand of Andrews.

We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we…

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…

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 Ap\'{e}ry numbers, by means of a general…

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…

We prove that if $q$ is a power of a prime $p$ and $p^k$ divides $a$, with $k\ge 0$, then \[ 1+(q-1)\sum_{0\le b(q-1)<a} \binom{a}{b(q-1)}\equiv 0\pmod{p^{k+1}}. \] The special case of this congruence where $q=p$ was proved by Carlitz in…

We prove that a certain family of sums of products of three binomials has alternating behavior modulo a prime $p$. To accomplish this we rewrite these sums as signed sums of products of three binomials, the better to handle $p$, and we give…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize…

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…

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…

Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a…

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n,…

Let $\lambda =\left( \lambda_{1},\lambda_{2},...,\lambda_{r}\right) $ be an integer partition, and $\left[p_{\lambda }\right] $ the $q$-analog of the symmetric power function $%p_{\lambda }$. This $q$-analogue has been defined as a special…

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…

We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/\pi^2$, and prove three of these examples.

A (p,q)-analogue of the classical Rogers-Szego polynomial is defined by replacing the q-binomial coefficient in it by the (p,q)-binomial coefficient. Exactly like the Rogers-Szego polynomial is associated with the q-oscillator algebra it is…

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3…

We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt--Clausen theorem and Kummer's…

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 $[n]\Phi_n(q)^3$. Here…