Relations between e, $π$, golden ratios and $\sqrt{2}$

TL;DR

This paper generalizes Euler's identity by introducing additive $p$-sequences and their associated $p$-golden ratios $\Phi_p$, defining $t_n(p)$ by $t_{n+p}=\sum_{k=0}^{p-1} t_{n+p-1-k}$ with seeds, and showing the limiting ratio $\Phi_p=\lim_{n\to\infty} t_{n+1}/t_n$ lies in $(1,2)$ and approaches $2$ as $p\to\infty$. It then defines the $p$-golden ratio via $X_p(x)=x^p-\sum_{k=1}^{p-1} x^k-1=0$, whose positive root is $\Phi_p$ and coincides with the long-run ratio of the $p$-sequence, e.g. $\Phi_2=(\sqrt{5}+1)/2$. A key result is the identity $e^{i\pi}+\Phi_p^p-\sum_{k=1}^{p-1} \Phi_p^k=0$ for $p\ge3$, derived from Euler's identity, along with various algebraic/trigonometric relations linking $e$, $\pi$, $\Phi_p$, and $\sqrt{2}$, including $\Phi=2\cos 36^{\circ}$ and related approximations. Overall, the work connects fundamental constants through generalized Fibonacci-type sequences and their limit ratios, extending Eulerian ideas to a broader family of constants.

Abstract

We write out relations between the base of natural logarithms ($e$), the ratio of the circumference of a circle to its diameter ($π$), the golden ratios ($Φ_p$) of the additive $p$-sequences, and the ratio of the diagonal of a square to its side ($\sqrt{2}$). An additive $p$-sequence is a natural extension of the Fibonacci sequence in which every term is the sum of $p$-previous terms given $p \ge 1$ initial values called seeds.

Figures (2)

• Figure 1: Division of a line into $2$ segments.
• Figure 2: Division of a line into $p$ segments.