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Title: Combinatorial identities and trigonometric inequalities
Authors: Alzer, H
Kwong, MK
Pan, H
Keywords: Combinatorial identities
Delannoy number
Fejer-Jackson inequality
Issue Date: 2016
Publisher: Polska Akademia Nauk, Instytut Matematyczny,Polish Academy of Sciences, Institute of Mathematics
Source: Colloquium mathematicum, 2016, v. 145, no. 2, p. 291-305 How to cite?
Journal: Colloquium mathematicum 
Abstract: The aim of this paper is threefold: (i) We offer short and elementary new proofs for (*) Sigma(n)(k=0)2(n-k)(n k) (m k) = Sigma(n)(k=0) (n k) (m+k k) (**) Sigma(n)(k=0) (alpha| k - 1 k) (z+1)l = alpha(alpha | n n) Sigma(n)(k=0) (n k ) z(k)/alpha+k icy k 1)(z + i)k (a+n)Vn (kn) zk 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) + Sigma(n)(k=2) c(c + 1) ... (c + k - 2) sin(kt)/k! > 0 (c is an element of R) or all n is an element of N and t is an element of (0, pi) if and only if c is an element of [-1,1]. This provides a new extension of the classical Fejer Jackson inequality.
ISSN: 0010-1354
EISSN: 1730-6302
DOI: 10.4064/cm6859-5-2016
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