Related papers: A binomial-coefficient identity arising from the m…
Based on Jensen formulae and the second kind of Chebyshev polynomials, another proof is presented for an extension of a curious binomial identity due to Z. W. Sun and K. J. Wu.
We show that Griffiths' multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group $\mathrm{SU}(1,d)$. Using this interpretation we derive several fundamental properties of the…
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…
We find the joint distribution of three simple statistics on lattice paths of n upsteps and n downsteps leading to a triple sum identity for the central binomial coefficient {2n}-choose-{n}. We explain why one of the constituent double sums…
The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study…
We propose an explicit representation of central $(2k+1)$-nomial coefficients in terms of finite sums over trigonometric constructs. The approach utilizes the diagonalization of circulant boolean matrices and is generalizable to all…
We present an elementary identity for the cyclotomic polynomials $\Phi_n(X)$ which reflects a kind of multiplicative property of $\Phi_n(X)$ as a function of $n$, and we explore its connections with the properties of other arithmetical…
Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of…
We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$.…
This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving $\zeta(2)$ and two other nice sums appear in the last section.
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…
This paper gives bijective proofs of some novel coinversion identities first discovered by Ayyer, Mandelshtam, and Martin (arxiv:2011.06117) as part of their proof of a new combinatorial formula for the modified Macdonald polynomials…
We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We…
We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…
In this paper, we prove two related central binomial series identities: $B(4)=\sum_{n \geq 0} \frac{\binom{2n}n}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $C(4)=\sum_{n \in \mathbb{N}} \frac{1}{n^4 \binom{2n}n}=\frac{17 \pi^4}{3240}.$ Both…
Let $s_{k}(n)$ denote the sum of digits of an integer $n$ in base $k$. Motivated by certain identities of Nieto, and Bateman and Bradley involving sums of the form $\sum_{i=0}^{2^{n}-1}(-1)^{s_{2}(i)}(x+i)^{m}$ for $m=n$ and $m=n+1$, we…
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers…
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…
In this paper we give a new formula for the $n$-th power of a $2\times2$ matrix. More precisely, we prove the following: Let $A= \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )$ be an arbitrary $2\times2$ matrix, $T=a+d$ its…