New Infinite Products of Cosines and Viète-Like Formulae

A complete view of Viète-like infinite products with Fibonacci and Lucas numbers

“…The possibility of doing this comes from the main result in [3] (namely, Theorem 1), which generalizes previous results in [1]. However, instead of just giving this general result, and looking for a well-balanced middle path between a self-contained and a purely expository paper, in this section we will provide a detailed exposition of the next pair of formulas in this process.…”

“…But instead of providing a detailed exposition of the results related to this case, we will just give the final result without proof as it is similar to the ðþ1; À1Þ case. Moreover, the analysis of the cycle ðÀ1; þ1Þ is treated in [1] in the simpler setting w 2 ½À2; 2, and this can be used as a guide to establish that for every positive even integer n, the pure imaginary i…”

“…The basic ingredients for deriving this result were new infinite products of cosines in conjunction with a formula that links nested square roots of 2 with certain cosines (see [1,2]). The main result in [4] reads ia n ffiffiffi 3 p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p q r Á Á Á ; ð1:3Þ…”

A Curious Trigonometric Infinite Product in Context

“…as a special case, originally obtained recursively by François Viète in 1593 and recently generalized by both Moreno and Garcia [9] and Levin [10], the latter of whose approach was based upon an an analysis of the functional equation…”