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

TL;DR

This paper proves a conjecture by Z. W. Sun involving a special sequence defined by binomial coefficients, providing analytic proofs for series related to the sequence and confirming a conjecture using series expansions of trigonometric functions.

Contribution

The paper offers the first analytic proofs of Sun's convergent series and confirms a conjecture using series expansions of sine and cosine functions.

Findings

Proved two convergent series involving the sequence S_n

Confirmed Sun's conjecture using series expansions of trigonometric functions

Established new identities related to binomial coefficient sequences

Abstract

Recently, Z. W. Sun introduced a sequence $(S_n)_{n\geq 0}$, where $S_n=\frac{\binom{6n}{3n} \binom{3n}{n}}{2(2n+1)\binom{2n}{n}}$, 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 $\cos(t\arcsin(x)).$