Proof of a conjecture of Z.-W. sun on trigonometric series

Abstract

Recently, Z. W. Sun introduced a sequence (Sn)n>o, where Sn =(gn 3n)(3n n)/2(2n+1) (2n n), and found one congruence and two convergent series on Sn by Mathematica. Furthermore, he proposed some related conjectures. In this paper, we first give analytic proofs of his two convergent series and then confirm one of his conjectures by invoking series expansions of sin(t arcsin(x)) and cos(t arcsin(x)).