Combinatorial identities and trigonometric inequalities

Horst Alzer ¹   ; Man Kam Kwong ²   ; Hao Pan ³

¹ Morsbacher Str. 10 51545 Waldbröl, Germany
² Department of Applied Mathematics The Hong Kong Polytechnic University Hunghom, Hong Kong
³ Department of Mathematics Nanjing University Nanjing 210093, People’s Republic of China

Colloquium Mathematicum, Tome 145 (2016) no. 2, pp. 291-305

Résumé

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.

Détail

DOI : 10.4064/cm6859-5-2016

Keywords: paper threefold offer short elementary proofs displaylines begin aligned * hskip sum n k biggl atop biggr biggl atop biggr sum biggl atop biggr biggl atop biggr ** hskip sum biggl alpha k atop biggr alpha biggl alpha atop biggr sum biggl atop biggr frac alpha end aligned first identity published brereton second extends result provided authors present q analogues * ** iii ** derive identities inequalities trigonometric polynomials among other results sin sum cdots k frac sin quad mathbb mathbb only provides extension classical fej jackson inequality