Transcendency of some constants related to integer sequences of polynomial iterations

Let $$P(x)=a_0x^d+a_1x^{d-1}+\cdots +a_d \in {{\mathbb {Q}}}[x]$$ be a polynomial of degree $$d \ge 2$$ , and let $$x_n$$ , $$n=0,1,2,\ldots $$ , be a sequence of integers satisfying $$x_{n+1}=P(x_n)$$ for $$n \ge 0$$ and $$x_n \rightarrow \infty $$ as $$n \rightarrow \infty $$ . Then, by a recent result of Wagner and Ziegler, $$\alpha =\lim _{n\rightarrow \infty } x_n^{d^{-n}}>1$$ is either an integer or an irrational number, and $$x_n$$ is approximately $$a_0^{-1/(d-1)} \alpha ^{d^n}-a_1/(da_0)$$ . Under assumption $$a_0^{1/(d-1)} \in {{\mathbb {Q}}}$$ on the leading coefficient $$a_0$$ of P, we completely characterize all the cases when the limit $$\alpha $$ is an algebraic number. Our results imply that $$\alpha $$ can be an integer, a quadratic Pisot unit with $$\alpha ^{-1}$$ being its conjugate over $${{\mathbb {Q}}}$$ , or a transcendental number. In most cases $$\alpha $$ is transcendental. For each $$d \ge 2$$ all the polynomials P of degree d for which $$\alpha $$ is an integer or a quadratic Pisot unit are described explicitly. The main theorem implies that several constants related to sequences that appear in a paper of Aho and Sloane and in the online Encyclopedia of Integer Sequences are transcendental.