Stirling’s Approximation for Central Extended Binomial Coefficients [PDF]
The American Mathematical Monthly, 2014 Slight modification of journal version; title ...
openaire +2 more sources
On certain congruences involving central binomial coefficients
Mathematica Montisnigri, 2023 Let p be an odd prime. In this paper, using some properties of Fibonacci numbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, we establish some congruences involving central binomial coefficient modulo p and p^2. We also give some new identities for hyperbolic functions.
Rachid Boumahdi +2 more
openaire +1 more source
Series associated with harmonic numbers, Fibonacci numbers and central binomial coefficients $binom{2n}{n}$ [PDF]
Notes on Number Theory and Discrete MathematicsWe find various series that involve the central binomial coefficients $binom{2n}{n}$, harmonic numbers and Fibonacci numbers. Contrary to the traditional hypergeometric function _pF_q approach, our method utilizes a straightforward transformation to ...
Segun Olofin Akerele +1 more
doaj +1 more source
Identities for squared central binomial coefficients
, 2021 We prove four identities for the squared central binomial coefficients. The first three of them reflect certain transformation properties of the complete elliptic integrals of the first and the second kind, while the last one is based on properties of the Lagrange polynomials.
openaire +3 more sources
On Divisibility of Convolutions of Central Binomial Coefficients [PDF]
The Electronic Journal of Combinatorics, 2014 Recently, Z. Sun proved that \[ 2(2m+1)\binom{2m}{m} \mid \binom{6m}{3m}\binom{3m}{m} \] for $m\in\mathbb{Z}_{>0}$. In this paper, we consider a generalization of this result by defining \[ b_{n,k}=\frac{2^{k}\, (n+2k-2)!!}{((n-2)!!\, k!}. \] In this notation, Sun's result may be expressed as $2\, (2m+1) \mid b_{(2m+1),(2m+1)-1}$ for $m\in\mathbb ...
openaire +2 more sources
The Overlooked Potential of Generalized Linear Models in Astronomy-III: Bayesian Negative Binomial Regression and Globular Cluster Populations [PDF]
, 2015 In this paper, the third in a series illustrating the power of generalized linear models (GLMs) for the astronomical community, we elucidate the potential of the class of GLMs which handles count data.
Buelens, B. +7 more
core +2 more sources
Products and Sums Divisible by Central Binomial Coefficients [PDF]
The Electronic Journal of Combinatorics, 2013 In this paper we study products and sums divisible by central binomial coefficients. We show that $$2(2n+1)\binom{2n}n\ \bigg|\ \binom{6n}{3n}\binom{3n}n\ \ \mbox{for all}\ n=1,2,3,\ldots.$$ Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k\ \bigg|\ \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k\ \bigg|\
openaire +3 more sources
A q-analog of Ljunggren's binomial congruence [PDF]
, 2011 We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that \[ \binom{a p}{b p} \equiv \binom{a}{b} \] modulo $p^3$ for primes $p\ge5$.
Straub, Armin
core +3 more sources
Some congruences involving binomial coefficients
, 2015 Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant ...
Cao, Hui-Qin, Sun, Zhi-Wei
core +1 more source
Harmonic Series of Convergence Ratio “1/4" with Cubic Central Binomial Coefficients
AxiomsWe examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “1/4”) containing the cubic central binomial coefficients and harmonic numbers is ...
Chunli Li, Wenchang Chu
doaj +1 more source