COMBINATORIAL IDENTITIES AND TRIGONOMETRIC INEQUALITIES

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.