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2013/2014 Remarks on a Sum involving Binomial Coefficients
Horst Alzer
Real Anal. Exchange 39(2): 363-366 (2013/2014).


Let \[ S_p(a,b;t)=\frac{1}{b}\sum_{k=0}^{p} \frac{{p\choose k}}{ {ak+b \choose b} } t^k, \] with \(p\in \mathbb{N}\), \(0<a\in\mathbb{R}\), \(0<b\in\mathbb{R}\), \( t\in\mathbb{R}\). We prove that \( S_p(a,b;t)\) is completely monotonic on \((0,\infty)\) as a function of \(a\) (if \(t>0\)) and as a function of \(b\) (if \(t\geq -1)\). This extends a result of Sofo, who proved that \(a\mapsto S_p(a,b;t)\) is strictly decreasing, convex, and log-convex on \([1,\infty)\).


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Horst Alzer. "Remarks on a Sum involving Binomial Coefficients." Real Anal. Exchange 39 (2) 363 - 366, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 30 June 2015

zbMATH: 1326.26022
MathSciNet: MR3365380

Primary: 11B65 , 26A48
Secondary: 11B65 , 26B20

Keywords: binomial coefficients , completely monotonic , Finite sum

Rights: Copyright © 2014 Michigan State University Press

Vol.39 • No. 2 • 2013/2014
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