相关论文: Proofs of some binomial identities using the metho…
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…
For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for…
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…
In this paper we prove some combinatorial identities which can be considered as generalizations and variations of remarkable Chu-Vandermonde identity. These identities are proved by using an elementary combinatorial-probabilistic approach…
We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.
We survey combinatorial interpretations of some dozen identities for the double factorial such as, for instance, (2n-2)!! + Sum_{k=2}^{n} (2n-1)!!(2k-4)!!/(2k-1)!! = (2n-1)!!. Our methods are mostly bijective.
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
Binomial versions of the Andrews-Gordon-Bressoud identities are given.
We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…
We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares…
With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $\rho,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^\rho \cdot (-4)^{\rho(n-1-k)} \equiv 0\pmod{2^{\rho-2}n\binom{2n}{n}}.$$
We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.
In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated…
We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the "simplest", in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments…
Combinatorial number system represents a non-negative natural numbers as sum of binomial coefficients. This paper presents an induction proof that there exists unique representation of every non-negative natural number $m$ as sum of $r$…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
We prove some combinatorial identities using the Polya urn and the closely related Hoppe urn.
We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open…
In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…
Brualdi and Ma found a connection between involutions of length $n$ with $k$ descents and symmetric $k\times k$ matrices with non-negative integer entries summing to $n$ and having no row or column of zeros. From their main theorem they…