On polyadic Liouville numbers

We study here polyadic Liouville numbers, which are involved in a series of recent papers.
The canonic expansion of a polyadic number $\lambda$ is of the form
$$\lambda= \sum_{n=0}^\infty a_{n} n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$
This series converges in any field of $p-$ adic numbers $\mathbb{\mathrm{Q}}_p$.
We call a polyadic number $\lambda$ a polyadic Liouville number, if for any $n$ and $P$ there exists a positive integer $A$ such that for all primes $p$, satisfying $p\leq P$ the inequality
$$\left|\lambda -A \right|_{p}<A^{-n}$$
holds.
Let $k\geq 2$ be a positive integer. We denote for a positive integer $m$
$$\Phi(k,m)=k^{k^{\ldots^{k}}}$$
Let
$$n_{m}=\Phi(k,m)$$
and let
$$\alpha=\sum_{m=0}^{\infty}(n_{m})!.$$
Theorem 1. For any positive integer $k\geq 2$ and any prime number $p$ the series $\alpha$ converges to a transcendental element of the ring $\mathbf{Z}_p.$ In other words, the polyadic number $\alpha$ is globally transcendental.