Proof of a Conjecture of Z.-W. Sun on Trigonometric Series

Abstract

Recently, Z. W. Sun introduced a sequence $(S_n{)}_{n\ge 0}$, where $S_n=\frac{\left(\genfrac{}{}{0ex}{}{6n}{3n}\right)\left(\genfrac{}{}{0ex}{}{3n}{n}\right)}{2(2n+1)\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}$, and found one congruence and two convergent series on $S_n$ by {\tt{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 $\cos (t\arcsin (x)).$

7 pages, to appear in Utilitas Math