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A new trigonometric identity with applications

Zhi-Wei Sun, Hao Pan

Abstract

In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.

A new trigonometric identity with applications

Abstract

In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer we prove that which is equivalent to the identity where stands for the th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer and positive integer we obtain where denotes the Bernoulli polynomial of degree .

Paper Structure

This paper contains 3 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1
  3. Proofs of Corollary \ref{['Cor1.1']} and Theorem \ref{['Th1.2']}

Key Result

Theorem 1.1

Let $n$ be any positive odd integer. Then, for any complex number $q$ with $|q|=1$ and $q^k\not=1$ for all $k=1,\ldots,n$, we have Equivalently, we have the trigonometric identity where $x$ is a real number, $U_m(z)$ is the $m$-th Chebyshev polynomial of the second kind, defined by $U_m(\cos\theta)=(\sin\, (m+1)\theta)/\sin\theta$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Lemma 2.1