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

Abstract

The main goal of this paper is to link the nth Fibonacci and Lucas numbers through certain infinite products of nested radicals. This work relies on recent results on Viète-like infinite products appeared in Moreno and García-Caballero (2013) [3]. We will analyze in detail one particular case of these formulas and we will show how our treatment covers and extends previous results in the literature.

Introduction

The first reported result which connects the nth Fibonacci number $F_n$ to the nth Lucas number $L_n$ through an infinite product of nested radicals appears in [7]. Specifically, T. J. Osler obtained two formulas which can be rewritten as

$$\frac{{\alpha }_n}{2n\ln \varphi }=\frac{\sqrt{2+{\beta }_n}}{2}\frac{\sqrt{2+\sqrt{2+{\beta }_n}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+{\beta }_n}}}}{2}\cdots \text{,}$$

where

$${\alpha }_n=\left\{\begin{array}{cc}\sqrt{5}{F}_n\hfill & \text{if}n\text{is even,}\hfill \\ L_n\hfill & \text{if}n\text{is odd,}\hfill \end{array}\right.{\beta }_n=\left\{\begin{array}{cc}{L}_n\hfill & \text{if}n\text{is even,}\hfill \\ \sqrt{5}{F}_n\hfill & \text{if}n\text{is odd.}\hfill \end{array}\right.$$

In deriving this result, Osler first adapted the standard procedure to deduce Viète’s infinite product for $\pi /2$ in order to consider the hyperbolic sine and cosine functions instead of the trigonometric ones. The only point remaining in Osler’s reasoning concerns the evaluation of $\sinh x$ and $\cosh x$ at the values $x=n\log \varphi$, with n a positive integer and $\varphi =(1+\sqrt{5})/2$ the Golden Ratio.

Some years later, a new formula bearing a striking resemblance with (1.1) was deduced by the authors in [4]. 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

$$\frac{i{\alpha }_n}{\sqrt{3}}=\sqrt{2-{\beta }_n}\sqrt{2-\sqrt{2-{\beta }_n}}\sqrt{2-\sqrt{2-\sqrt{2-{\beta }_n}}}\cdots \text{,}$$

where i stands for the imaginary unit, n is a positive integer, and ${\alpha }_n$ and ${\beta }_n$ are defined as in (1.2). Here and in what follows, $\sqrt{·}$ denotes the principal value of the complex square root function. Therefore, if z is a nonzero complex number, then $\sqrt{z}=\sqrt{|z|}{e}^{i\frac{\text{Arg}(z)}{2}}$, where $|·|$ is the modulus function and $\text{Arg}(·)$ is the principal value of the argument function, which takes values in $(-\pi \text{,}\pi ]$.

Perhaps the most remarkable difference between formulas (1.1) and (1.3) is the sign inside the corresponding nested radicals. Only plus signs for the nested square roots in (1.1), and only minus signs for the ones in (1.3). Some interesting questions arise from this observation. What happens for different choices of signs inside the nested square roots? In other words, for a given sequence of signs, is it possible to get a closed form expression? This question, in its full generality, is far from being solved. But if we restrict our attention to the case in which the signs inside the nested square roots form a periodic pattern, then we can give a complete answer. For example, we will show that for even n, a closed form expression for

$$\left(\sqrt{2-L_n}\sqrt{2+\sqrt{2-L_n}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{2-L_n}}}\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2-L_n}}}}\right)\cdots$$

reads $i\sqrt{2+\varphi }{F}_n$. As the reader may have noticed, the signs in (1.4), when written from right to left, alternate starting with the minus sign. Also, for convergence purposes, and starting from the left, we must group each two consecutive factors in the infinite product.

Our purpose is to develop a general procedure to obtain a complete set of formulas such as (1.1), (1.3) and (1.4). 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.

The structure of the paper is the following: In Section 2 we will determine the two formulas with alternating signs; the proofs will be similar in spirit to the ones in [4]. 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.