# A Combinatorial Proof of An Identity Involving Trigonometric Power Sums

## Abstract

Let $n,m,d,r$ be integers where $n,m,d&space;\geq&space;1$. In this paper, we use a combinatorial proof to show that

$\sum_{k=1}^{d-1}\cos\left&space;(&space;\frac{(nm+2-2r)k\pi}{d}&space;\right&space;)\sin^n\left&space;(&space;\frac{mk\pi}{d}&space;\right&space;)\csc^n\left&space;(&space;\frac{k\pi}{d}&space;\right&space;)=-m^n+d\sum_{k=\left&space;\lceil&space;\frac{n-r}{d}&space;\right&space;\rceil}^{\left&space;\lfloor&space;\frac{mn-r}{d}&space;\right&space;\rfloor}\sum_{j=0}^{\left&space;\lfloor&space;\frac{dk+r-n}{m}&space;\right&space;\rfloor}(-1)^j&space;\binom{n}{j}\binom{dk+r-mj-1}{n-1}$

by counting the number of solutions for the congruence $x_1+x_2+\cdots+x_n\equiv&space;r\pmod{d}$ where $1\leq&space;x_1,x_2,\ldots,x_n&space;\leq&space;m$.

## Article Details

How to Cite
Kaikeaw, R., & Naenudorn, K. (2022). A Combinatorial Proof of An Identity Involving Trigonometric Power Sums. Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 67(708), 25–39. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/245989
Section
Research Article

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