2019/08/06 by Zhi‐Wei Sun, Sun, Zhi-Wei
Mathematics · Physics and Astronomy · #05A19 #11A15 #33B10 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1908.02155
openalex publication_date 2019/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we obtain some novel identities involving trigonometric functions. Let n be any positive odd integer. We show that ∑r=0n-1\frac11+sin2π\fracx+rn+cos2π\fracx+rn =\frac(-1)(n-1)/2n1+(-1)(n-1)/2sin 2πx+cos 2πx for any complex number with x+1/2,x+(-1)(n-1)/2/4\not∈\mathbb Z, and ∑j,k=0n-1\frac1sin 2π\fracx+jn+sin2π\fracy+kn=\frac(-1)(n-1)/2n2sin 2πx+sin2πy for all complex numbers x and y with x+y,x-y-1/2\not∈\mathbb Z. We also determine the values of ∏k=1(p-1)/2(1+tanπ\frack2p) and ∏k=1(p-1)/2(1+\cotπ\frack2p) for any odd prime p. In addition, we pose several conjectures on the values of Gp(x)=∏k=1(p-1)/2(x-e2πik2/p) with p an odd prime and x a root of unity.