The plastic number and its generalized polynomial

Abstract

The polynomial $X^3-X-1$ has a unique positive root known as plastic number, which is denoted by $\rho$ and is approximately equal to $1.32471795$. In this note we study the zeroes of the generalized polynomial $X^k-{\sum }_{j=0}^{k-2}{X}^j$ for $k\ge 3$ and prove that its unique positive root ${\lambda }_k$ tends to the golden ratio $\varphi =\frac{1+\sqrt{5}}{2}$ as $k\to \infty$. We also derive bounds on ${\lambda }_k$ in terms of Fibonacci numbers.

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References in corpus (2)

• Three Series for the Generalized Golden Mean
• Limits of zeros of polynomial sequences