On Viète-like formulas

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

“…It is worth pointing out that ðða rÞ=ð2 N r À 1ÞÞðp=2Þ, the argument of the secant function in formula (3.2), is always in the interval ð0; p=2Þ, as proved in [3,Proposition 2]. Moreover, the only case for which ða rÞ=ð2 N r À 1Þ ¼ 1 is the one corresponding to the choice b 1 ¼ b 2 ¼ Á Á Á ¼ b N ¼ 1, which does not fulfill the requirements of Theorem 3.1.…”

“…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.…”

“…For this purpose, we provide a detailed exposition of the main result in [3], which generalizes formula (2.3) and Theorem 2.5.…”

“…The assumption to be made is that the signs inside the nested radicals must follow some periodic pattern, in a precise sense that will be determined below. To achieve our aim we will adapt the main result in [3], and show that both (1.3) and (1.4) are particular cases of our general description. Also, we will expose why Osler's formula (1.1) should be considered as the spurious case of the given family of formulas.…”

“…In Section 3 we provide a detailed exposition of Theorem 1 in [3], which is the core of our general formulation. And the last section contains our main result, with previous results from the literature reviewed in this more general setting.…”