By means of the hypergeometric series approach, we present a new proof of Sun’s conjecture on trigonometric series, which is simpler than the original one due to Sun and Meng. Several further infinite series identities are shown as examples. | Turk J Math (2018) 42: 2935 – 2941 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Trigonometric expressions for infinite series involving binomial coefficients Nadia N. LI∗ School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, . China Received: • Accepted/Published Online: • Final Version: Abstract: By means of the hypergeometric series approach, we present a new proof of Sun’s conjecture on trigonometric series, which is simpler than the original one due to Sun and Meng. Several further infinite series identities are shown as examples. Key words: Binomial series, hypergeometric series, trigonometric functions 1. Introduction and motivation For an integer n and an indeterminate x , define the rising and falling factorials, respectively, by the following quotients of Euler’s Γ -function: (x)n = Γ(x + n) Γ(x) and ⟨x⟩n = Γ(1 + x) , Γ(1 + x − n) where for the former we shall utilize the abbreviated multiparameter notation below: [ A, B, α, β, ··· , ··· , C γ ] = n (A)n (B)n · · · (C)n . (α)n (β)n · · · (γ)n According to Bailey [1, §], the classical hypergeometric series reads as [ 1+p Fp ∞ ] ∑ (a0 )k (a1 )k (a2 )k · · · (ap )k k a0 , a1 , a2 , · · · , ap z . z = b1 , b 2 , · · · , b p k!(b1 )k (b2 )k · · · (bp )k k=0 By introducing the integer sequence (6n)(3n) Sn = 3n (2n) , n 2(2n + 1) n Sun [6] proposed the following conjecture. Conjecture 1 There are positive integers T1 , T2 , T3 . such that √ ∞ ∞ ∑ ∑ cos ( 23 arccos (6 3x)) 1 − Tk x2k + Sk x2k+1 = 24 12 k=1 k=0 ∗Correspondence: lina2017@ 2010 AMS Mathematics Subject Classification: Primary 33C20, Secondary 05A10 2935 This work is licensed under a Creative Commons Attribution International License. LI/Turk J Math √ for all real x with |x| ≤ 1/6 3. Also, Tp ≡ −2 (mod p) for any prime p . Sun and Meng [5] gave an analytic proof for .
