Recursive Properties of Trigonometric Products

Russell Jay Hendel and Charles K. Cook

Chapter 17 in Applications of Fibonacci Numbers, 1996, pp 201-214 from Springer

Abstract: Abstract A variety of authors — Lind [17,18], Zeitlin [27], Swamy [25], Sjogren [24], Bruckman [4], Cooper-Kennedy [7] and Shapiro [23] — have presented identities equating the values of finite products involving trigonometric functions with members of sequences satisfying second order recursions. Some examples are: (1) $$ {F_n} = \prod\limits_{k = 1}^{\left[ {\frac{{n - 1}}{2}} \right]} {\left( {3 + 2\cos \left( {\frac{{2\pi k}}{n}} \right)} \right)} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n = 1,2,3 \ldots $$ (2) $$ {P_n} = {2^{\left[ {\frac{n}{2}} \right]}}\prod\limits_{k = 1}^{\left[ {\frac{{n - 1}}{2}} \right]} {\left( {3 + \cos \left( {\frac{{2\pi k}}{n}} \right)} \right),\,\,\,\,\,\,\,\,\,\,\,n = 1,2,3 \ldots } $$ (3) $$ {L_n} = \prod\limits_{k = 0}^{\left[ {\frac{{n - 2}}{2}} \right]} {\left( {3 + 2\cos \left( {\frac{{\left( {2k + 1} \right)\pi }}{n}} \right)} \right)} ,\,\,\,\,\,\,n = 2,3,4 \ldots . $$

Keywords: Closed Formula; Proof Method; Hypergeometric Equation; Polynomial Sequence; Polynomial Family (search for similar items in EconPapers)
Date: 1996
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DOI: 10.1007/978-94-009-0223-7_17

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