scholarly journals An algebraic identity on q-Apéry numbers

2011 ◽  
Vol 311 (23-24) ◽  
pp. 2708-2710 ◽  
Author(s):  
De-Yin Zheng
Keyword(s):  
Algebraic Identity ◽  
Apéry Numbers

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

scholarly journals Another congruence for the Apéry numbers

1987 ◽  
Vol 25 (2) ◽  
pp. 201-210 ◽  
Author(s):  
F. Beukers
Keyword(s):  
Apéry Numbers

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

A serendipitous path to a famous inequality

2008 ◽  
Vol 92 (523) ◽  
pp. 50-54
Author(s):  
Robert M. Young
Keyword(s):  
Real Number ◽  
Geometric Means ◽  
Algebraic Identity ◽  
Binomial Expansion

The mysterious path of discovery – the tireless experimentation in search of patterns, the veiled connections that suddenly unfold, serendipity – all these elements combine to make mathematics so magical. The purpose of this note is to show how a routine algebraic identity, the binomial expansion of (x- 1)2, can be used to give a new proof of the fundamental inequality between the arithmetic and geometric means. The proof will provide further evidence that a great deal of useful mathematics can be derived from the obvious assertion that the square of a real number is never negative.

scholarly journals Hankel-type determinants for some combinatorial sequences

2018 ◽  
Vol 14 (05) ◽  
pp. 1265-1277 ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Zhi-Wei Sun
Keyword(s):  
Nonnegative Integer ◽  
Apéry Numbers

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer [Formula: see text], define [Formula: see text] [Formula: see text] For [Formula: see text], we show that [Formula: see text] and [Formula: see text] are positive odd integers, and [Formula: see text] and [Formula: see text] are always integers.

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