Arithmetic properties of Euler-type series with a Liouvillian polyadic parameter

This paper states that, for any nonzero linear form $h_0f_0(1)+h_1f_1(1)$ with integer coefficients $h_0,h_1$, there exist infinitely many $p$-adic fields where this form does not vanish. Here, $f_0(1)=\sum\limits_{n=0}^\infty (\lambda)_n$, $f_1(1)=\sum\limits_{n=0}^\infty(\lambda+1)_n$, $\lambda$ where $\lambda$ is a Liouvillian polyadic number and $(\lambda)_n$ stands for the Pochhammer symbol. This result shows the possibility of studying the arithmetic properties of values of hypergeometric series with transcendental parameters.