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Combinatorial identities and trigonometric inequalities

Volume 145 / 2016

Horst Alzer, Man Kam Kwong, Hao Pan Colloquium Mathematicum 145 (2016), 291-305 MSC: Primary 05A19; Secondary 05A30, 26D05, 30C10, 42A05. DOI: 10.4064/cm6859-5-2016 Published online: 8 July 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.

Authors

  • Horst AlzerMorsbacher Str. 10
    51545 Waldbröl, Germany
    e-mail
  • Man Kam KwongDepartment of Applied Mathematics
    The Hong Kong Polytechnic University
    Hunghom, Hong Kong
    e-mail
  • Hao PanDepartment of Mathematics
    Nanjing University
    Nanjing 210093, People’s Republic of China
    e-mail

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