Inequalities for binomial coefficients
type: Item
instance of: scholarly article
title: Inequalities for binomial coefficients
zbMATH Open document ID: 0941.05002
author: Zoltán Sasvári
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-03T00:00:00Z
review text: For any real number \(r\) with \(r>1\), let \(c_r= (2\pi(1-{1\over r}))^{-1/2}\) and \(d_r= (r-1)/(1-{1\over r})^r\). Let \(B_{2m}\) \((m= 1,2,\dots)\) be the Bernoulli numbers defined by \[ {z\over e^z-1}=1-{z\over 2}+\sum^\infty_{m=1} B_{2m}{z^{2m}\over (2m)!}. \] The author proves that if \(s\) is a positive number, then \[ c_r d^s_r e^{D_{2k+ 1}(s,r)}/\sqrt s< {rs\choose s}< c_r d^s_r e^{D_{2k}(s,r)}/\sqrt s \] for \(k= 0,1,2,\dots\), where \[ D_n(s,r)= \sum^n_{j=1} B_{2j}((rs)^{1- 2j}- s^{1- 2j}- ((r- 1)s)^{1- 2j})/2j(2j- 1) \] for \(n= 0,1,2,\dots\).
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 1340108
zbMATH Keywords: binomial coefficients
zbMATH Keywords: Bernoulli numbers
reviewed by: Le Maohua
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6420
OpenAlex ID: W1986018265
cites work: Q5645344
cites work: An Elementary Proof of Binet's Formula for the Gamma Function
cites work: Q3844537
DOI: 10.1006/JMAA.1999.6420
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description: scientific article; zbMATH DE number 1340108