Abstract
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 \([1,\infty)\).
Citation
Horst Alzer. "Remarks on a Sum involving Binomial Coefficients." Real Anal. Exchange 39 (2) 363 - 366, 2013/2014.
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