Related papers: Demonstration de la conjecture de Dumont
We give a generalization and a short mechanized proof of determinant conjectured by G. Kuperberg and J. Propp. Further generalizations and applications of the method to some q-analogues may be found in http://www.math.temple.edu/~tewodros
It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…
Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…
We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L. K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we construct a bijection through different combinatorial objects,…
While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some…
Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…
We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of q-binomial theorem and a…
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and…
We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by…
Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding…
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to…
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…
We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the…
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula…
We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity,…
In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that…
We provide a simple proof of a generalization of the multivariate Chu-Vandermonde identity recently derived in Favaro et al. (2010a). Exploiting known results for rising factorials and fourth Lauricella polynomials we show resorting to…
In this paper, we employ the theories and techniques of hypergeometric functions to provide two distinct proofs of the conjectured identities involving multiple Ap\'ery-like series with central binomial coefficients and multiple harmonic…