The symbol \({\text{or}}{{{\text{d}}}_{p}}\alpha \) denotes the power to which the prime number p is raised in the canonical decomposition of a rational number α. The p-adic norm of a rational number \(\alpha \) is defined by the equality |α|p = \({{p}^{{ - {\text{or}}{{{\text{d}}}_{p}}\alpha }}}\). The field \({{\mathbb{Q}}_{p}}~\) of p-adic numbers is the completion of the rational number field with respect to the p-adic norm. The ring of integer polyadic numbers is the direct product of the rings of integer p-adic numbers over all prime numbers p. The canonical representation of an element θ of the ring of integer polyadic numbers has the form of a series θ = \(\mathop \sum \limits_{n = 0}^\infty {{a}_{n}}n!\), \({{a}_{n}} \in \mathbb{Z}\), \(0 \leqslant {{a}_{n}} \leqslant n.\) This series converges in all fields of p-adic numbers. Therefore, it can be treated as an infinite-dimensional vector, whose coordinates in the corresponding ring of integer p-adic numbers are denoted by θ(p).

The foundations of the theory of polyadic numbers are described in [1]. Note that this theory finds applications in the theory of Abelian groups. Moreover, some applications make use of the so-called factorial decomposition of positive integers (it is rather short), which represents a partial sum of the series θ.

A polyadic number θ is called a polyadic Liouville number (or a Liouville polyadic number) if for any numbers \(n\) and P there exists a positive integer A such that, for all prime numbers p satisfying the inequality \(p \leqslant P~\) it is true that \({{\left| {\theta - A} \right|}_{p}} < {{A}^{{ - n}}}\). A polyadic Liouville number is a transcendental element of any field of p‑adic numbers.

An important area in the theory of transcendental numbers is the study of the arithmetic nature of the values of generalized hypergeometric series, i.e., series of the form

$$\mathop \sum \limits_{n = 0}^\infty \frac{{{{{\left( {{{\alpha }_{1}}} \right)}}_{n}} \ldots {{{\left( {{{\alpha }_{r}}} \right)}}_{n}}}}{{{{{\left( {{{\beta }_{1}}} \right)}}_{n}} \ldots {{{\left( {{{\beta }_{s}}} \right)}}_{n}}}}{{z}^{n}},$$

where \({{\left( \gamma \right)}_{n}}\) is the Pochhammer symbol defined by the equalities \({{\left( \gamma \right)}_{0}} = 1,\) \({{\left( \gamma \right)}_{n}} = \gamma \left( {\gamma + 1} \right) \ldots \left( {\gamma + n - 1} \right)~\) for \(n \geqslant 1.\) If such series have rational parameters, then they are reduced to Siegel E- or G-functions or to F‑series. As a result, it is possible to apply the Siegel–Shidlovskii method from the theory of transcendental numbers and its modification (see, e.g., [2–8]). This short list does not purport to be complete, but it gives an idea of the character of the main results.

If the parameters include algebraic irrational numbers, then the arithmetic properties of the series can be studied using Hermite–Padé approximations (see, e.g., [9–11]).

In all the works mentioned above, the parameters of series are algebraic numbers. In this paper, we describe a new approach for studying the arithmetic nature of the values of generalized hypergeometric series of the form

$$\mathop \sum \limits_{n = 0}^\infty {{\left( {{{\alpha }_{1}}} \right)}_{n}} \ldots {{\left( {{{\alpha }_{m}}} \right)}_{n}}{{z}^{n}},$$
(1)

with parameters \({{\alpha }_{1}}, \ldots ,{{\alpha }_{m}}\) being transcendental polyadic Liouville numbers. Special cases of this problem for the series

$$\begin{gathered} \mathop \sum \limits_{n = 0}^\infty {{\left( \lambda \right)}_{n}},\quad \mathop \sum \limits_{n = 0}^\infty {{\left( {\lambda + 1} \right)}_{n}}, \\ \mathop \sum \limits_{n = 0}^\infty {{\left( \lambda \right)}_{n}}{{\lambda }^{n}},\quad ~\mathop \sum \limits_{n = 0}^\infty {{\left( {\lambda + 1} \right)}_{n}}{{\lambda }^{n}} \\ \end{gathered} $$

with a Liouvillian polyadic parameter λ were considered in [12–14]. All these works, including the present one, rely heavily on the Hermite–Padé approximations from [15].

Definition. 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 exist infinitely many prime numbers p such that \({{h}_{1}}\theta _{1}^{{\left( p \right)}} + \ldots + {{h}_{m}}\theta _{m}^{{\left( p \right)}} \ne 0\) in the field \({{\mathbb{Q}}_{p}}~\).

Additionally, of interest are problems dealing with prime numbers only from some proper subsets of the set of prime numbers. In this case, we talk about infinite linear independence with constraints on the indicated set. Let M be a positive integer. Consider a reduced system of residues mod(M). As usual, the number of elements of this system is denoted by φ(M), where φ(M) is the Euler function. All prime numbers belong to the union of arithmetic progressions with a common difference equal to M and with the first terms forming a reduced system of residues mod(M). Let \({{a}_{1}}\),…,\({{a}_{\rho }}\) be ρ distinct elements chosen arbitrarily from this system. Denote by \({{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}\) the sets of positive integer values taken by the progressions \({{a}_{i}}\) + Mk, \(i = 1,\,\, \ldots ,\,\,\rho ,\,\,~k \in \mathbb{Z}.\) Let \({\mathbf{P}}~\left( {{{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}} \right).\) denote the set of prime numbers contained in the union of the sets \({{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}.\) The theorems given below state that the values of generalized hypergeometric series with constraints on the set \({\mathbf{P}}~\left( {{{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}} \right).\) are infinitely linearly independent. This approach was earlier used in [16] as applied to the Euler series \(\mathop \sum \limits_{n = 0}^\infty n!{{\left( { - z} \right)}^{n}}.\)

Note once again that the goal of this study is to obtain results concerning the values of series (1) with parameters including transcendental polyadic Liouville numbers.

Now we formulate the main results of this work. Let m be a positive integer such that \(m \geqslant 2\). Let λ0 be an arbitrary positive integer greater than 1. Define \({{s}_{0}} = \left[ {{\text{exp}}{{\lambda }_{0}}} \right] + 1\). Let λ1 be an arbitrary positive integer satisfying the following condition: for any prime number \(p \leqslant {{s}_{0}} + 2\lambda _{0}^{2},\) it is true that \({\text{or}}{{{\text{d}}}_{p}}{{\lambda }_{1}} \geqslant m{{s}_{0}}{\text{ln}}{{s}_{0}}~\), and let \({{s}_{1}}\, = \,[{\text{exp}}{{\lambda }_{1}}]\, + \,1\). For \(k \geqslant 1~\), let \({{\lambda }_{{k + 1}}}\) be an arbitrary positive integer satisfying the following condition: for any prime number \(p\, \leqslant \,{{s}_{k}}\, + \,2\lambda _{k}^{2}\), it is true that \({\text{or}}{{{\text{d}}}_{p}}{{\lambda }_{{k + 1}}}\, \geqslant \,m{{s}_{k}}{\text{ln}}{{s}_{k}}\), and let \({{s}_{{k + 1}}} = \left[ {{\text{exp}}{{\lambda }_{{k + 1}}}} \right] + 1\). Let \({{\mu }_{{i,0,~}}},\,\,i = 1,\,\, \ldots ,\,\,m\), be positive integers. Let \({{\mu }_{{i,k}}}\) for any \(i = 1,\,\, \ldots ,\,\,m,\,\,k \geqslant 1\), be nonnegative integers satisfying the inequality \({{\mu }_{{i,k}}}\, \leqslant \,{{\lambda }_{k}}.\)

Let \({{\alpha }_{i}} = \mathop \sum \limits_{l = 0}^\infty {{\mu }_{{i,l}}}{{\lambda }_{l}},~\,\,i = 1,\,\, \ldots ,\,\,m.~\) If there is k such that \({{\mu }_{{i,l}}} = 0\) for all \(l \geqslant k\), then αi is a positive integer. Otherwise, this series is a polyadic Liouville number.

Consider the series

$$\begin{gathered} {{f}_{0}}\left( z \right) = \mathop \sum \limits_{n = 0}^\infty {{\left( {{{\alpha }_{1}}} \right)}_{n}} \ldots {{\left( {{{\alpha }_{m}}} \right)}_{n}}{{z}^{n}}, \\ {{f}_{i}}\left( z \right) = \mathop \sum \limits_{n = 0}^\infty {{\left( {{{\alpha }_{1}} + 1} \right)}_{n}} \ldots {{\left( {{{\alpha }_{i}} + 1} \right)}_{n}}{{\left( {{{\alpha }_{{i + 1}}}} \right)}_{n}} \ldots {{\left( {{{\alpha }_{m}}} \right)}_{n}}{{z}^{n}}, \\ i = 1,\,\, \ldots ,\,\,m - 1, \\ \end{gathered} $$
$${{f}_{m}}\left( z \right) = \mathop \sum \limits_{n = 0}^\infty {{\left( {{{\alpha }_{1}} + 1} \right)}_{n}} \ldots {{\left( {{{\alpha }_{m}} + 1} \right)}_{n}}{{z}^{n}}.$$

Theorem 1. Let \(m \geqslant 2,M\) and ρ be positive integers. Suppose that \(~\left( {m + 1} \right)\rho \) \( > \varphi (M)m.\)Then, for any integers \({{h}_{0}},\,\, \ldots ,\,\,{{h}_{m}},\) that are not all zero and for any positive integer ξ, there exist infinitely many prime numbers p from the set \({\mathbf{P}}~\left( {{{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}} \right)\) such that in the field \({{\mathbb{Q}}_{p}}\)

$${{\left| {L\left( \xi \right)} \right|}_{p}} = {{\left| {{{h}_{0}}{{f}_{0}}\left( \xi \right) + \ldots + {{h}_{m}}{{f}_{m}}\left( \xi \right)} \right|}_{p}} > 0.$$

Let μk be positive integers satisfying the inequality \({{\mu }_{k}} \leqslant \) \({{\lambda }_{k}}\) for any k. Define Ξ = \(\mathop \sum \limits_{l = 0}^\infty {{\mu }_{l}}{{\lambda }_{l}}.\)

Theorem 2. Let \(~m \geqslant 2,\) \(M,\) and ρ be positive integers. Assume that \(~\left( {m + 1} \right)\rho \) \( > \varphi (M)m.\) Then, for any integers \({{h}_{0}},\,\, \ldots ,\,\,{{h}_{m}}\) that are not all zero, there exist infinitely many prime numbers \(p\) from the set \({\mathbf{P}}~\left( {{{{\mathbf{a}}}_{1}},\,\, \ldots ,\,\,{{{\mathbf{a}}}_{\rho }}} \right)\) such that in the field \({{\mathbb{Q}}_{p}}\)

$${{\left| {L\left( {{\Xi }} \right)} \right|}_{p}} = {{\left| {{{h}_{0}}{{f}_{0}}\left( {{\Xi }} \right) + \ldots + {{h}_{m}}{{f}_{m}}\left( {{\Xi }} \right)} \right|}_{p}} > 0.$$

In the inequalities of the claims of the theorems, the symbols \({{f}_{i}}\left( \xi \right),~\,\,{{f}_{i}}\left( {{\Xi }} \right),\,\,i = 0,\,\, \ldots ,\,\,m,\) denote the sums of these series in \({{\mathbb{Q}}_{p}}\).

It is possible to conjecture that the values of the series considered in Theorems 1 and 2 are linearly independent in any field of p-adic numbers and the value of any series considered in these theorems is a transcendental number in any field of p-adic numbers. Under certain natural conditions on the parameters, it is natural to conjecture that the values of these series are algebraically independent in any field of p-adic numbers.

Such results still cannot be obtained by applying methods available in the theory of transcendental numbers in a polyadic domain (the state of the art in this issue can be found in [8]). Nevertheless, we intend to use the ideas of this work to develop a generalized Siegel–Shidlovskii method and to obtain general theorems similar to those from [8], but for the class of series with transcendental coefficients. The infinite algebraic independence of the values of series of form (1) with transcendental parameters would be proved by applying the approaches from [3, 4].

Another important direction of future research is to study the statistical properties of digits of factorial decompositions.