Recursive Properties of Trigonometric Products

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:

$$ {F_n} = \prod\limits_{k = 1}^{\left[ {\frac{{n - 1}}{2}} \right]} {\left( {3 + 2\cos \left( {\frac{{2\pi k}}{n}} \right)} \right)} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n = 1,2,3 \ldots $$

((1))

$$ {P_n} = {2^{\left[ {\frac{n}{2}} \right]}}\prod\limits_{k = 1}^{\left[ {\frac{{n - 1}}{2}} \right]} {\left( {3 + \cos \left( {\frac{{2\pi k}}{n}} \right)} \right),\,\,\,\,\,\,\,\,\,\,\,n = 1,2,3 \ldots } $$

((2))

$$ {L_n} = \prod\limits_{k = 0}^{\left[ {\frac{{n - 2}}{2}} \right]} {\left( {3 + 2\cos \left( {\frac{{\left( {2k + 1} \right)\pi }}{n}} \right)} \right)} ,\,\,\,\,\,\,n = 2,3,4 \ldots . $$

((3))