Infinite linear independence with constraints on a subset of prime numbers of values of Eulerian-type series with polyadic Liouville parameter

A ring of polyadic integers is a direct product of rings of integer $p$-adic numbers over all primes $p$. The elements $\theta$ of this ring can thus be considered as infinite-dimensional vectors whose coordinates in the corresponding ring of integer $p$-adic numbers are denoted by $\theta^{(p)}$. The infinite linear independence of polyadic numbers $\theta_{1},\ldots,\theta_{m}$ means that for any nonzero linear form $h_{1}x_{1}+\ldots+h_{m}x_{m}$ with integer coefficients $h_{1},\ldots,h_{m}$ there is an infinite set of primes $p$ such that in the field $\mathbb{\mathrm{Q}}_p$ the inequality
$$h_{1}\theta_{1}^{(p)}+\ldots+h_{m}\theta_{m}^{(p)}\neq 0$$
holds. At the same time, problems in which primes are considered only from some proper subsets of the set of primes are of interest. In this case, we will talk about infinite linear independence with restrictions on the specified set. Canonical representation of the element $\theta$ of the ring of polyadic integers has the form of a series
$$ \theta= \sum_{n=0}^\infty a_{n} n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$
Of course, a series whose members are integers converging in all fields of $p$-adic numbers is a polyadic integer. We will call a polyadic number $\theta$ a polyadic Liouville number (or a Liouville polyadic number) if for any numbers $n$ and $P$ there exists a natural number $A$ such that for all primes $p$ satisfying the inequality $p\leq P$ the inequality
$$\left|\theta-A\right|_{p}<A^{-n}.$$
This work continues the development of the basic idea embedded in [Chirskiy_ist15]. Here the infinite linear independence with restrictions on the set of prime numbers in the aggregate of arithmetic progressions. of polyadic numbers
$$ f_{0}(1)=\sum_{n=0}^\infty (\lambda)_{n}, f_{1}(1)=\sum_{n=0}^\infty (\lambda +1)_{n}.$$
will be proved. An important apparatus for obtaining this result are Hermite–Pade approximations of generalized hypergeometric functions constructed in the work of Yu.V. Nesterenko [4]. The approach from the work of Ernvall-Hytonen, Matala-Aho, Seppela [5] was used.