FINITE TRIGONOMETRIC PRODUCT AND SUM IDENTITIES. 1. Introduction In some recent papers [1, 2, 4, 5, 6], one finds product identities such as.
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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 ad 1 Itroductio I some recet papers 1,, 4, 5, 6], oe fids product idetities such as 1/ 1/ cos 3+cos ] 1, odd 11 ] 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 xe irπ/ y, r1 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 1+y 1y ] y 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 13 holds sice both sides have degree 1 resp for odd resp eve, share zeros at ±ita/ for s 1,, 1/, ad evaluate to oe at y 1 Evaluatig this formula at several values gives various idetities Sometimes these may be simplified if oe uses a double-agle formula 13 AUGUST 01 17
2 THE FIBONACCI QUARTERLY y 5 : y i : coefficiet of domiat term : coefficiet of ext domiat term : y 0 : y 3i : coefficiet of y : y 3 : 3+cos cos cos ] F ] ] /, eve, 1, odd 0, 0 mod 4, 1 1/4, 1 mod 4, 1 /4, mod 4, 1 3/4, 3 mod 4 ta /11/3, eve, 1/, odd si 1 1cos cot 5+4cos ] 1 6 0, 0,3 mod 6, 1, 1, mod 6, 1, 4,5 mod 6 ] ] These idetities may be foud or derived from formulas i Hase 3] Specifically, the secod formula correspods to 913, the third to 91, the fourth to 11, the fifth to 9114, the sixth to 919, ad the seveth to 301 Note that the right side of the eighth formula is always a iteger Differetiatig 13 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 VOLUME 50, NUMBER 3
3 FINITE TRIGONOMETRIC PRODUCT AND SUM IDENTITIES The secod formula relates to Hase s 611 ad 61 The followig polyomial equatio ivolves the ta fuctio ad a sum of odd powers: ] 1+x ta 1 1+x +1x ] 15 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 ] 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 9133 Differetiatig 15 gives xta 1+x ta 1+x q 1x q q +1 1+x +1x 16 AUGUST 01 19
4 THE FIBONACCI QUARTERLY 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] 17 x 5 : x 1 : x i : coefficiet of domiat term : coefficiet of ext domiat term : x 3i : ] 1 4 ta ] 1 cos 1 1 ] 1 ta 1 ta 1 1 cot qq ] 1 4 ta L q 0, q odd, 1 q/, q eve 1/, q 1, mod 3, 1, q 0 mod 3 The secod formula relates to Hase s 916, the fourth formula with 9135, ad the fifth formula with VOLUME 50, NUMBER 3
5 Differetiatig 17 gives This produces the special cases x 5 : FINITE TRIGONOMETRIC PRODUCT AND SUM IDENTITIES 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 18 ta 1 1+5ta 1 qf q1 L q 1 si q ta 1 q 1ta 1, q eve 1 ta q 1q ta 1 13ta 1 The third formula relates to Hase s 114 Refereces q/4, q 0 mod 3, q/, q 1 mod 3, 0, q mod 3 1] N Cahill, J R D Errico, ad J Spece, Complex factorizatios of the Fiboacci ad Lucas umbers, The Fiboacci Quarterly , ] N Garier ad O Ramaré, Fiboacci umbers ad trigoometric idetities, The Fiboacci Quarterly 46/ /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 NS, , MSC010: 11B39, 33B10 Departmet of Mathematics ad Statistics, Griell College, Griell, IA address: chamberl@mathgrielledu AUGUST 01 1
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