FINITE TRIGONOMETRIC PRODUCT AND SUM IDENTITIES. In some recent papers [1], [2], [4], [5], [6], one finds product identities such as. 2cos.

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1 FINITE TRIGONOMETRIC PRODUCT AND SUM IDENTITIES MARC CHAMBERLAND Abstract. Several product ad sum idetities are established with special cases ivolvig Fiboacci ad Lucas umbers. These idetities are derived from polyomial idetities ispired by the Biet formulas for Fiboacci ad Lucas umbers. I some recet papers 1], ], 4], 5], 6], oe fids product idetities such as 1/ cos ] 1, odd 1.1 ad 1/ 3+cos ] F, odd 1. where F is the th Fiboacci umber. The goal of this ote is to uify these ad other beautiful product or sum formulas as special istaces of three polyomial idetities. While the three idetities are special istaces of the geeral equatio x y r1 xe irπ/ y, the ispiratio for the specialized cases arises from the familiar Biet formulas for Fiboacci umbers F ad Lucas umbers L. This approach cotrasts with those of previously cited papers which use lesser-kow represetatios of the Fiboacci umbers. The first formula is 1+y 1cos ] 1 +y 1y ] y 1.3 for each atural umber. To prove that the polyomials o each side of the equatio are equal, it suffices to show that they have the same degree, the same zeros, ad evaluate to the same o-zero value at oe poit. Formula 1.3 holds sice both sides have degree 1 resp. for odd resp. eve, share zeros at ±ita/ for s 1,...,, ad evaluate to oe at y 1. Evaluatig this formula at several values gives various idetities. 1

2 MARC CHAMBERLAND Sometimes these may be simplified if oe uses a double-agle formula. y ] 5 : 3+cos F 0, 0 mod 4 ] 1 1/4, 1 mod 4 y i : cos 1 /4, mod 4 1 3/4, 3 mod 4 coefficiet of domiat term : coefficiet of ext domiat term : y 0 : y 3i : coefficiet of y : y 3 : cos ] /, eve, 1, odd ta /11/3, eve, 1/, odd si 1 1cos ] cot cos ] ] 0, 0,3 mod 6 1, 1, mod 6 1, 4,5 mod 6 These idetities may be foud or derived from formulas i Hase3]. Specifically, the secod formula correspods to 91..3, the third to 91.., the fourth to 1.1., the fifth to , the sixth to 91..9, ad the seveth to Note that the right side of the eighth formula is always a iteger. Differetiatig 1.3 gives Specific choices yield 1 y + ycos 1+y 1cos y 5 : 1+10 y i : y 3i : cos 3+cos sec cos 1 4cos 1+y 1+y L 1 F 1 + 1y 1y 1 1/, 1 mod 4 0, mod 4 1/, 3 mod 4 1/3, 1 mod 3 /6, mod 3 1.4

3 PRODUCT AND SUM IDENTITIES 3 The secod formula relates to Hase s ad The followig polyomial equatio ivolves the ta fuctio ad a sum of odd powers: ] 1+x ta 1 1+x +1x ]. 1.5 q +1 This idetity is prove with the same approach as before: both sides have degree q, share zeros at x ±icot/q +1 for s 1,...,q, ad evaluate to oe whe x 0. Special choices iclude x 5 : x 1 : x i : coefficiet of domiat term : coefficiet of ext domiat term : x 3i : ] 4 ta L q +1 ] 1, q 0,3 mod 4 cos q +1 1, q 1, mod 4 1 ] 1, q 0,3 mod 4 ta q +1 1, q 1, mod 4 q +1, q 0,3 mod 4 ta q +1 q +1, q 1, mod 4 cot qq +1 q ] 4 ta q +1 1, q 0, mod 3, q 1 mod 3 The fourth formula relates to Hase Differetiatig 1.5 gives xta 1+x ta 1+x q 1x q q +1 1+x +1x 1.6

4 4 MARC CHAMBERLAND This produces the special cases x 5 : x 1 : x i : coefficiet of x : x 3i : ta 1+5ta q +1F q 4L si q +1 q +1 4 ta 1ta ta q +1q q +1 ta 13ta, q eve 1, q odd 0, q 0 mod 3 q +1/8, q 1 mod 3 q +1/4, q mod 3 The last polyomial equatio is similar to the secod, but with eve powers: Special choices iclude ] 1 1+x ta 1 1+x q +1x q]. 1.7 x 5 : x 1 : x i : coefficiet of domiat term : coefficiet of ext domiat term : x 3i : ] 1 4 ta L q ] 1 cos 1 ] 1 ta 1 ta 1 1 cot qq ] 1 4 ta 0, q odd, 1 q/, q eve, 1/, q 1, mod 3 1, q 0 mod 3 The secod formula relates to Hase s 91..6, the fourth formula with , ad the fifth formula with

5 Differetiatig 1.7 gives This produces the special cases x 5 : PRODUCT AND SUM IDENTITIES 5 xta 1 1+x ta 1 x 1 : x i : coefficiet of x : x 3i : q 1+xq1 1x q1 1+x q +1x q 1.8 ta 1 1+5ta 1 1 si ta 1 1ta 1 qf q1 L q q q,q eve 1 ta q 1q ta 1 13ta 1 The third formula relates to Hase s Refereces q/4, q 0 mod 3 q/, q 1 mod 3 0, q mod 3 1] N. Cahill, J.R. D Errico, J. Spece: Complex factorizatios of the Fiboacci ad Lucas umbers. Fiboacci Quarterly , ] N. Garier ad O. Ramaré: Fiboacci Numbers ad Trigoometric Idetities. Fiboacci Quarterly 46/47 008/09, ] E. Hase: A Table of Series ad Products. Pretice-Hall, Eglewood Cliffs, ] J. Seibert ad P. Trojovsky: Circulats ad the factorizatio of the Fiboacci-like umbers. Acta Math. Uiv. Ostrav , ] B. Sury: Of grad-auts ad Fiboacci. Mathematical Gazette 9 008, ] B. Sury: Trigoometric expressios for Fiboacci ad Lucas umbers. Acta Math. Uiv. Comeia. N.S , MSC010: 11B39, 33B10 Departmet of Mathematics ad Statistics, Griell College, Griell, IA address: chamberl@math.griell.edu