Combinatorial identities and trigonometric inequalities
Volume 145 / 2016
Abstract
The aim of this paper is threefold: (i) We offer short and elementary new proofs for $$\displaylines{\begin{aligned} (*)\hskip82pt\sum_{k=0}^{n} 2^{n-k} \biggl({n\atop k}\biggr)\biggl({m\atop k}\biggr) ={}&\sum_{k=0}^n \biggl({n\atop k}\biggr)\biggl({m+k\atop k}\biggr),\\ (**)\hskip55pt \sum_{k=0}^n \biggl({\alpha+k-1\atop k}\biggr)(z+1)^k={}& \alpha \biggl({\alpha+n\atop n}\biggr)\sum_{k=0}^n\biggl({n\atop k}\biggr)\frac{z^k}{\alpha+k}. \end{aligned} } $$ The first identity was published by Brereton et al. in 2011 and the second one extends a result provided by the same authors. (ii) We present $q$-analogues of $(*)$ and $(**)$. (iii) We use $(**)$ to derive identities and inequalities for trigonometric polynomials. Among other results, we show that $$ \sin(t)+ \sum_{k=2}^n c (c+1) \cdots (c+k-2) \frac{\sin(kt)}{k! } \gt 0 \quad\ {(c\in\mathbb{R})} $$ for all $n\in\mathbb{N}$ and $t\in (0,\pi)$ if and only if $c\in [-1,1]$. This provides a new extension of the classical Fejér–Jackson inequality.
