A Combinatorial Proof of An Identity Involving Trigonometric Power Sums

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A combinatorial proof of an identity involving trigonometric power sums
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Published: Dec 28, 2022
Keywords:
Trigonometric power sums, Trigonometric series, Combinatorial proof, Inclusion-Exclusion principle, Generating functions

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Rachanai Kaikeaw
Faculty of Science and Technology, Dhonburi Rajabhat University, Bangkok 10600
Khanithar Naenudorn
Faculty of Education, Dhonburi Rajabhat University, Bangkok 10600

Abstract

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


gif.latex?\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 gif.latex?x_1+x_2+\cdots+x_n\equiv&space;r\pmod{d} where gif.latex?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
Issue
Vol. 67 No. 708 (2022): September - December 2022
Section
Research Article

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