论文标题
与多项式递归相关的生长常数的非理性性
Irrationality of growth constants associated with polynomial recursions
论文作者
论文摘要
我们考虑满足表单$ x_ {n+1} = p（x_n）$的整数序列，用于某些多项式$ p $ a $ d> 1 $。如果这样的序列倾向于无穷大，那么它可以满足$ x_n \ simaα^{d^n} $的渐近公式，但是关于常数$α$的渐近公式。在本文中，我们表明$α$始终是不合理的或整数。实际上，我们证明了一个更强有力的陈述：如果序列$ g_n $满足$ g_n =aα^n + b + o（α^{ - εn}）$的渐近公式，其中$ a，b $是代数是$ a> $ a> 1 $，而序列则包含了无限的整体$ nirgention $ is nir niste nir an Ante ant an nir an An An An An An An An An An An An An An Ante。
We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A α^{d^n}$, but little can be said about the constant $α$. In this paper, we show that $α$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A α^n + B + O(α^{-εn})$, where $A,B$ are algebraic and $α> 1$, and the sequence contains infinitely many integers, then $α$ is irrational or an integer.
